Calculus / English

Partial derivatives, introduction

Khan Academy · YouTube · 10:55

YouTube · Khan AcademyOriginal video

Selected moment · 0:15 · Chapter markers are time references, not playback progress.

IN THIS VIDEO

01Scalar-valued multivariable function example
02Partial derivatives as the multivariable analogue of ordinary derivatives
03Leibniz notation for an ordinary derivative

FROM THE LEARNING NOTES

f(x,y)=x^2y+\sin(y)

Scalar-valued multivariable function example

SELECTED MOMENT

Why partial derivatives are needed

SUGGESTED PREPARATION

Ordinary single-variable derivatives
Function notation
Basic Cartesian graphs
Concept of input and output spaces
Functions of several variables
Ordinary derivatives of single-variable functions
Reading mathematical notation such as $f(x,y)$ and evaluating at a point
Basic single-variable differentiation
Power rule
Derivative of $\sin(y)$ is $\cos(y)$
Derivative of a constant is $0$
Comfort with function notation and substitution
Basic trigonometric derivatives (e.g., $d/dx \sin (x) = \cos (x)$ )
Concept of functions of multiple variables

READ & KEEP

The explanation, unpacked.

Reviewed learning material · Video analysis · English

Read the full overview

This introductory whiteboard segment explains partial derivatives by first revisiting ordinary derivatives. It starts with a scalar-valued two-variable function $f(x,y)=x^2y+\sin(y)$ , then uses $f(x)=x^2$ and the notation $\frac{df}{dx}(2)$ to explain derivatives in two ways: as local slope on a graph, with $dx$ and $df$ shown as tiny input and output changes, and as a mapping between input and output number lines. The speaker then transfers this intuition to the multivariable setting, writing $\frac{\partial f}{\partial x}$ and interpreting it as the effect of a tiny change in the $x$ direction on the output. Finally, the input space is redrawn as the $xy$ -plane, emphasizing that each point in the plane is an input rather than a point on the graph of the function, and the example point $(1,2)$ is marked before the clip ends mid-explanation. This 180-second introductory calculus segment explains partial derivatives for a two-variable function. Using $f(x,y)=x^2y+\sin(y)$ , the presenter first visualizes how a change in $x$ or $y$ separately affects the scalar output, then introduces the notation $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ to emphasize multivariable dependence. The clip contrasts this with a single-variable analogy, erases that comparison panel, and begins a worked setup for $\frac{\partial f}{\partial x}(1,2)$ by treating $y$ as constant and rewriting the problem as an ordinary derivative in $x$ . The final numerical evaluation is not reached within the provided duration. This 180-second whiteboard segment introduces partial derivatives through the concrete function $f(x,y)=x^2y+\sin(y)$ . It first computes the two partial derivatives at the point $(1,2)$ , obtaining $\frac{\partial f}{\partial x}(1,2)=4$ and $\frac{\partial f}{\partial y}(1,2)=1+\cos(2)$ . The speaker then shifts from pointwise evaluation to a more general viewpoint, explaining that one often wants a formula valid for any $(x,y)$ . By treating the other variable as constant symbolically rather than numerically, the clip derives $\frac{\partial f}{\partial x}(x,y)=2xy+0$ , i.e. $2xy$ , and notes that substituting $(1,2)$ recovers the earlier numerical answer. This video introduces partial derivatives by demonstrating how to compute them for a function of two variables, $f(x,y) = x^2y + \sin (y)$ . It explains that calculating a partial derivative involves treating all other variables as constants and then applying standard single-variable differentiation rules. The geometric interpretation is also discussed, describing partial derivatives as measuring the rate of change when 'nudging' the input in one specific direction. Finally, it highlights that while graphs and slopes are intuitive for simple cases, a more general understanding based on input-output ratios is necessary for higher dimensions or vector-valued functions.

Use the learning inspector for key ideas and moments, or open the reading tabs for the complete notes.

Chapters

0:00A scalar-valued multivariable function 0:15Why partial derivatives are needed 0:30Review of ordinary derivative notation 0:45Derivative as slope on a graph 1:35Derivative as a number-line mapping 2:15From ordinary derivatives to partial derivatives 2:35The

xy

-plane as input space 3:00Directional changes of a multivariable function 4:10Introducing partial-derivative notation 4:50Clearing the one-dimensional analogy 5:10Setting up

\frac{\partial f}{\partial x}(1,2)

6:00Compute

\frac{\partial f}{\partial x}(1,2)

6:20Compute

\frac{\partial f}{\partial y}(1,2)

7:20From point values to a general formula 8:00Derive

\frac{\partial f}{\partial x}(x,y)

9:00Introduction to Calculating Partial Derivatives 10:05General Method and Geometric Interpretation 10:25Beyond Graphs: A More General View of Derivatives

Learning script

Generated from the video's visuals and explanation; not verbatim speech.

The segment opens by introducing a concrete two-variable function, written on the board as $f(x,y)=x^2y+\sin(y)$ . The speaker identifies it as scalar-valued, meaning that although the input has two coordinates, the output is a single number.

That setup raises the central question of the clip: how do we differentiate an expression with more than one input variable? The answer given is the method of partial derivatives, which the speaker immediately frames as closely related to ordinary derivatives rather than as a completely separate idea.

To build that connection, the video first reviews the single-variable case. The board shows $f(x)=x^2$ and the Leibniz notation $\frac{df}{dx}(2)$ , signaling that the discussion will focus on what the derivative symbol means when evaluated at a particular input point.

A graph of $f(x)=x^2$ is then drawn. The speaker interprets $dx$ as a tiny horizontal nudge in the input direction and $df$ as the resulting vertical change in the output. In this graphical picture, the derivative is the local slope, the rise-over-run ratio produced by those two small changes at the chosen point $x=2$ .

Next, the same idea is restated without a graph. The input space and output space are drawn as two number lines, and the function is viewed as mapping points from one line to the other. Under this interpretation, the derivative measures how strongly a small input displacement is amplified into an output displacement; the speaker illustrates this with the example that an output nudge four times as large corresponds to derivative value 4 at that point.

With that intuition established, the clip returns to the multivariable setting. The notation $\frac{\partial f}{\partial x}$ is written, and the speaker explains it as asking how a tiny change in the input specifically in the $x$ direction influences the output of a function of several variables.

Finally, the visualization changes from a one-dimensional input line to a two-dimensional input space. An $xy$ -plane is drawn, and the speaker stresses that this plane is not the graph of the function; instead, every point in the plane is an input pair. The example point $(1,2)$ is marked to begin the same kind of local-change reasoning in the multivariable context, though the clip ends before that explanation is completed.

The segment opens with the example function $f(x,y)=x^2y+\sin(y)$ already on the board. The speaker asks how changing one input affects the output, and the drawing makes the situation concrete: the input lives on an $xy$ -plane, while the output is represented as a separate number line labeled $f$ .

A yellow horizontal arrow labeled $dx$ is added on the input plane. The curved arrow from the plane to the number line expresses the function as a map, and the corresponding arrow on the output line is labeled $df$ . Mathematically, this is the idea of measuring the output response to a perturbation in the $x$ -direction only.

The same reasoning is then repeated for the second variable. A red vertical arrow labeled $dy$ is drawn on the input plane, and a different output displacement is marked on the number line. The point is that $f$ can respond differently when the input moves in the $y$ -direction than when it moves in the $x$ -direction.

The speaker writes the informal expressions $\frac{df}{dx}(1,2)$ and $\frac{df}{dy}(1,2)$ , indicating that both directional sensitivities are being considered at the same input point $(1,2)$ . This keeps the focus on local change in one coordinate at a time.

Next, the notation is refined. The board changes from ordinary $d$ notation to $\frac{\partial f}{\partial x}(1,2)$ and $\frac{\partial f}{\partial y}(1,2)$ . The speaker explains that the curled symbol $\partial$ is used to signal that the function is multivariable and that the derivative records only one part of the full behavior of $f$ .

This leads to an important conceptual caution: neither $\frac{\partial f}{\partial x}$ nor $\frac{\partial f}{\partial y}$ tells the whole story of how $f$ changes. Each one isolates a single coordinate direction, so each is only a partial description of the function’s local sensitivity.

The right-hand comparison panel showing the one-variable function $f(x)=x^2$ and $\frac{df}{dx}(2)$ is erased. That visual contrast has served its purpose, and the lesson now turns from analogy to direct computation in the multivariable example.

The speaker rewrites the target quantity at the top right as $\frac{\partial f}{\partial x}(1,2)$ . The key rule is stated plainly: when differentiating with respect to $x$ , the variable $y$ is treated as constant. At the point $(1,2)$ , that means $y$ can be replaced by $2$ before any differentiation is done.

The board then shows the transformed expression $\left.\frac{\partial}{\partial x}\left(x^2\cdot 2+\sin(2)\right)\right|_{x=1}$ . This is the central computational move of the clip: the multivariable partial-derivative question has been converted into a single-variable derivative problem in $x$ , followed by evaluation at $x=1$ .

The speaker notes that what remains is “just an ordinary derivative,” because after substituting $y=2$ the expression depends only on $x$ . The clip stops at this setup stage, before carrying out the derivative and producing the final numerical value.

The clip opens on a prepared whiteboard with the title “Partial derivatives” and the function $f(x,y)=x^2y+\sin(y)$ already written. The immediate task is to evaluate $\frac{\partial f}{\partial x}(1,2)$ . The speaker substitutes $y=2$ first, producing $\frac{\partial}{\partial x}(x^2\cdot 2+\sin(2))\big|_{x=1}$ , then differentiates term by term: $x^2\cdot 2$ becomes $4x$ , while $\sin(2)$ is constant with respect to $x$ and contributes $0$ . Finally, setting $x=1$ gives the numerical answer $4$ .

Next the speaker switches to the other coordinate direction and computes $\frac{\partial f}{\partial y}(1,2)$ . Here the fixed variable is $x=1$ , so the expression is rewritten as $\frac{\partial}{\partial y}((1)^2y+\sin(y))\big|_{y=2}$ . Differentiating with respect to $y$ gives $1+\cos(y)$ , and then substituting $y=2$ yields $1+\cos(2)$ . The speaker explicitly leaves the trigonometric value unevaluated numerically, emphasizing the exact symbolic form of the answer.

After finishing the two pointwise examples, the lecture pivots to a broader question: often one does not want just the derivative at a single point, but a general rule that works for any input $(x,y)$ . The board is partially cleared to make room for this new goal. The key conceptual shift is that the same “hold the other variable constant” principle still applies, but now the constant is represented symbolically rather than by a number.

The speaker then derives the general $x$ -partial directly from $f(x,y)=x^2y+\sin(y)$ , writing $\frac{\partial f}{\partial x}(x,y)=\frac{\partial}{\partial x}(x^2y+\sin(y))$ . Treating $y$ as a constant multiplier, $x^2y$ differentiates to $2xy$ ; treating $\sin(y)$ as a constant term, its derivative is $0$ . The resulting formula on screen is $2xy+0$ , equivalent to $2xy$ . The clip closes by noting that plugging in $(1,2)$ to this general formula reproduces the earlier pointwise result $4$ .

Let's calculate the partial derivative of our function $f(x,y) = x^2y + \sin (y)$ with respect to y, denoted as ∂ $f/y$ . We set up the expression: ∂/∂ $y (x^2y + \sin (y))$ .

When differentiating with respect to y, we treat x as a constant. Therefore, $x^2$ is a constant coefficient for y. The derivative of a constant times y is just the constant itself, so ∂/∂ $y(x^2y) = x^2$ .

For the term $\sin (y)$ , there are no x's involved, so we simply take its ordinary derivative with respect to y, which is $\cos (y)$ . Combining these, we get ∂f/∂ $y = x^2 + \cos (y)$ .

This result is a general formula for the partial derivative. If we were to evaluate this at the point (1, 2), we would substitute $x=1$ and $y=2$ , yielding $1^2 + \cos (2) = 1 + \cos (2)$ . Note: The speaker initially says 'cosine of 1', but based on the point (1,2) and the formula, it should be 'cosine of 2'.

This process demonstrates the core method for computing partial derivatives: pretend all other variables are constants, then take an ordinary derivative. Conceptually, this corresponds to moving or 'nudging' the input in one specific direction (e.g., along the y-axis) and observing how the output changes.

While graphs and slopes provide a helpful intuition for simple cases, they aren't the only way to understand derivatives. For functions with higher-dimensional inputs or vector-valued outputs, graphical representations become insufficient.

In such cases, a more general perspective is valuable: view the derivative as the ratio of an output 'nudge' to an input 'nudge' in a chosen direction. This foundational idea will be crucial for understanding more advanced topics in multivariable calculus.

Knowledge cards

Scalar-valued multivariable function example

The clip begins with a concrete function of two variables, $f(x,y)=x^2y+\sin(y)$ . The speaker explicitly notes that it takes a two-variable input but outputs a single number, so it is scalar-valued. This example motivates the need for a differentiation method suited to functions with more than one input.

f(x,y)=x^2y+\sin(y)

Partial derivatives as the multivariable analogue of ordinary derivatives

After asking how to differentiate a multivariable expression, the speaker introduces partial derivatives and says they are very similar to ordinary derivatives. The purpose of the segment is to show that the same basic input-change/output-change intuition behind ordinary derivatives carries over to the multivariable case.

Leibniz notation for an ordinary derivative

Before discussing partial derivatives, the video reviews the single-variable notation $\frac{df}{dx}$ using the example $f(x)=x^2$ evaluated at $x=2$ . This notation is chosen because it makes the upcoming interpretation in terms of small changes explicit.

\frac{df}{dx}(2)

Graphical meaning of $dx$ and $df$

On the graph of $f(x)=x^2$ , the speaker interprets $dx$ as a tiny nudge in the input direction and $df$ as the resulting change in the output. Their ratio is then identified with the slope of the graph at the chosen point, giving a local geometric meaning to the derivative.

\frac{df}{dx}=\text{slope}

Derivative as amplification between input and output number lines

The same derivative idea is reformulated without graphs. The input space and output space are treated as separate number lines, and the function maps one to the other. In this view, the derivative tells you how much a small input displacement is magnified into an output displacement; the speaker illustrates this with an example where the output change is four times the input change.

Meaning of $\frac{\partial f}{\partial x}$

The clip transfers the previous intuition to functions of several variables. The notation $\frac{\partial f}{\partial x}$ is explained as measuring how a tiny change in the input in the $x$ direction influences the output. This is the core conceptual definition of the partial derivative presented in the segment.

\frac{\partial f}{\partial x}

For two-variable functions, the input space is the $xy$ -plane

When moving from one variable to two variables, the speaker draws an $xy$ -plane and emphasizes that it represents input space, not the graph of the function. Each point in the plane corresponds to an input pair such as $(1,2)$ , and partial differentiation is then discussed relative to such points in the domain.

Partial derivative as change in one input direction

For a function such as $f(x,y)=x^2y+\sin(y)$ , a partial derivative measures how the output changes when only one input coordinate is varied. The video visualizes this by drawing a perturbation in the input plane and showing the induced change on the output number line.

$dx$ , $dy$ , and $df$ in the visual model

$dx$ denotes a small change in the $x$ -direction, $dy$ denotes a small change in the $y$ -direction, and $df$ denotes the resulting change in the function value. The diagrams distinguish the two input directions by using separate arrows on the plane and separate output displacements on the number line.

dx,\ dy,\ df

Why the symbol $\partial$ is used

The notation changes from $\frac{df}{dx}$ and $\frac{df}{dy}$ to $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ to emphasize that the function has multiple variables. The curled symbol signals a partial derivative rather than an ordinary single-variable derivative.

\frac{\partial f}{\partial x},\quad \frac{\partial f}{\partial y}

Each partial derivative is only part of the story

A single partial derivative does not describe all possible changes of a multivariable function. $\frac{\partial f}{\partial x}$ tracks only motion in the $x$ -direction, and $\frac{\partial f}{\partial y}$ tracks only motion in the $y$ -direction.

Computing $\frac{\partial f}{\partial x}$ by freezing $y$

To evaluate $\frac{\partial f}{\partial x}(1,2)$ , the video treats $y$ as a constant and substitutes $y=2$ first. This reduces the problem to differentiating an expression that depends only on $x$ , then evaluating at $x=1$ .

\frac{\partial f}{\partial x}(1,2)=\left.\frac{\partial}{\partial x}\left(x^2\cdot 2+\sin(2)\right)\right|_{x=1}

Partial derivative at a point

For a function of several variables, a partial derivative at a point is found by freezing all but one coordinate and differentiating in the remaining direction. In this clip, $\frac{\partial f}{\partial x}(1,2)$ is computed by first fixing $y=2$ , and $\frac{\partial f}{\partial y}(1,2)$ by first fixing $x=1$ . The outputs are numbers attached to the specific point $(1,2)$ .

\frac{\partial f}{\partial x}(1,2),\quad \frac{\partial f}{\partial y}(1,2)

Worked value of $\frac{\partial f}{\partial x}(1,2)$

Starting from $f(x,y)=x^2y+\sin(y)$ , substitute $y=2$ to get $x^2\cdot 2+\sin(2)$ . Differentiate with respect to $x$ : the derivative of $x^2\cdot 2$ is $4x$ , and the derivative of the constant $\sin(2)$ is $0$ . Evaluating at $x=1$ gives $4$ .

\frac{\partial f}{\partial x}(1,2)=\frac{\partial}{\partial x}(x^2\cdot 2+\sin(2))\Big|_{x=1}=4x+0\Big|_{x=1}=4

Worked value of $\frac{\partial f}{\partial y}(1,2)$

Now fix $x=1$ instead. The function becomes $(1)^2y+\sin(y)=y+\sin(y)$ . Differentiating with respect to $y$ gives $1+\cos(y)$ , and evaluating at $y=2$ yields $1+\cos(2)$ . The speaker intentionally leaves the cosine unevaluated numerically.

\frac{\partial f}{\partial y}(1,2)=\frac{\partial}{\partial y}((1)^2y+\sin(y))\Big|_{y=2}=1+\cos(y)\Big|_{y=2}=1+\cos(2)

General partial derivative formula

A general partial derivative is a function of the input variables rather than a single number. To obtain it, do not substitute numerical constants first; instead, treat the non-differentiated variable symbolically as a constant. This produces a rule that can later be evaluated at any point.

\frac{\partial f}{\partial x}(x,y)

Deriving $\frac{\partial f}{\partial x}(x,y)$ for the example

Apply the same constant-holding logic to $f(x,y)=x^2y+\sin(y)$ without plugging in numbers. Since $y$ is treated as constant, $x^2y$ differentiates to $2xy$ , and $\sin(y)$ differentiates to $0$ . Thus the general formula is $2xy+0$ , i.e. $2xy$ . Substituting $(1,2)$ recovers the earlier answer $4$ .

\frac{\partial f}{\partial x}(x,y)=\frac{\partial}{\partial x}(x^2y+\sin(y))=2xy+0

Common pitfall: forgetting which variable is constant

The main method emphasized throughout the clip is that a partial derivative changes only one coordinate direction. When computing $\frac{\partial f}{\partial x}$ , treat $y$ as constant; when computing $\frac{\partial f}{\partial y}$ , treat $x$ as constant. Mixing this up changes the problem from a partial derivative to an ordinary one-variable derivative of the wrong expression.

Definition of Partial Derivative

A partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant. To compute ∂ $f/x$ , treat y as a constant and differentiate normally. To compute ∂f/∂y, treat x as a constant and differentiate normally.

\frac{\partial f}{\partial x}, \quad \frac{\partial f}{\partial y}

Calculating ∂ $f/y$ for $f(x,y) = x^2y + \sin (y)$

To find the partial derivative with respect to y, treat x as a constant. The derivative of $x^2y$ with respect to y is $x^2$ (since $x^2$ is constant). The derivative of $\sin (y)$ with respect to y is $\cos (y)$ . Thus, ∂f/∂ $y = x^2 + \cos (y)$ .

\frac{\partial}{\partial y}(x^2y + \sin(y)) = x^2 + \cos(y)

Geometric Meaning of Partial Derivatives

Partial derivatives represent the rate of change of the function's output when the input is 'nudged' in the direction of one specific variable, while keeping the other variable(s) fixed. It measures the sensitivity of the output to changes in a single input dimension.

Limitations of Graphical Interpretation

While graphs and slopes are intuitive for simple functions, they cannot fully represent derivatives for vector-valued functions or functions with inputs of higher dimensions than two. A more general understanding involves viewing derivatives as the ratio of an output change to an input change in a specific direction.

Detailed learning notes

Explore conditions, steps and evidence. Supplementary explanations are labeled separately from content shown in the video.

Symbols · 39

$f(x,y)$

Clear evidence

Shown in the video

Evidence

Audio
Observation
The speaker says, "So let's say I have some multivariable function like f of x y" and then describes it as having a two-variable input.
Formula
Observation
The board shows the handwritten expression $f(x,y)$ .

Symbol

$f(x,y)$

Meaning

A multivariable function with two inputs, x and y.

Domain

Two-variable input space; in this example the output is a single number.

x

Clear evidence

Shown in the video

Evidence

Formula
Observation
The expression $f(x,y)=x^2y+\sin(y)$ is written on the board.

Symbol

Meaning

One of the two independent input variables of the multivariable function.

Domain

Input variable of $f(x,y)$ .

y

Clear evidence

Shown in the video

Evidence

Formula
Observation
The expression $f(x,y)=x^2y+\sin(y)$ is written on the board.

Symbol

Meaning

The second independent input variable of the multivariable function.

Domain

Input variable of $f(x,y)$ .

$f(x)$

Clear evidence

Shown in the video

Evidence

Audio
Observation
The speaker says, "So if you have something like f of x is equal to x squared".
Formula
Observation
The board shows $f(x)=x^2$ .

Symbol

$f(x)$

Meaning

An ordinary single-variable function used as a comparison case for partial derivatives.

Domain

Single-variable input space.

$\frac{df}{dx}$

Clear evidence

Shown in the video

Evidence

Audio
Observation
The speaker says, "and I'll use the Leibniz notation here, df dx".
Formula
Observation
The board shows $\frac{df}{dx}$ .

Symbol

$\frac{df}{dx}$

Meaning

Leibniz notation for the derivative of $f$ with respect to $x$ .

Domain

Ordinary derivative notation for a single-variable function.

dx

Clear evidence

Shown in the video

Evidence

Audio
Observation
The speaker says, "This little dx here, I like to interpret as just a little nudge in the x direction".
Diagram
Observation
A small horizontal arrow labeled $dx$ is drawn near $x=2$ on the graph.

Symbol

Meaning

A tiny change or "nudge" in the input direction $x$ .

Domain

Infinitesimal-style input increment in the visual interpretation of the derivative.

df

Clear evidence

Shown in the video

Evidence

Audio
Observation
The speaker says, "And then df ... is the resulting change in the output after you make that initial little nudge".
Diagram
Observation
A vertical arrow labeled $df$ is drawn above the point at $x=2$ .

Symbol

Meaning

The resulting change in the output caused by the input change $dx$ .

Domain

Infinitesimal-style output increment in the visual interpretation of the derivative.

$x=2$

Clear evidence

Shown in the video

Evidence

Audio
Observation
The speaker says, "let's evaluate it at 2" and later "over here we have x equals 2".
Formula
Observation
The board shows $\frac{df}{dx}(2)$ .
Diagram
Observation
The graph marks the input location $x=2$ .

Symbol

$x=2$

Meaning

The specific input point at which the ordinary derivative is being evaluated in the example.

Domain

Point in the input space of $f(x)=x^2$ .

$\frac{\partial f}{\partial x}$

Clear evidence

Shown in the video

Evidence

Audio
Observation
The speaker says, "You know, you could write df dx and interpret that as saying, hey, how does a tiny change in the input in the x direction influence the output?" while discussing the multivariable setting.
Formula
Observation
The board shows $\frac{\partial f}{\partial x}$ .

Uncertainties

The spoken wording uses "df dx," but the displayed formula is the partial-derivative symbol $\frac{\partial f}{\partial x}$ .

Symbol

$\frac{\partial f}{\partial x}$

Meaning

Partial derivative notation indicating how a tiny change in the input in the $x$ direction influences the output of a multivariable function.

Domain

Multivariable function setting; here introduced for $f(x,y)$ .

(1,2)

Clear evidence

Shown in the video

Evidence

Audio
Observation
The speaker says, "let's say you were evaluating this at a point like 1, 2".
Formula
Observation
The board shows $(1,2)$ next to $\frac{\partial f}{\partial x}$ .
Diagram
Observation
A point is marked on the $xy$ -plane corresponding to the input $(1,2)$ .

Symbol

(1,2)

Meaning

A specific input point in the $xy$ -plane at which the partial derivative is being considered.

Domain

Input space of the multivariable function $f(x,y)$ .

xy\text{-plane}

Clear evidence

Shown in the video

Evidence

Audio
Observation
The speaker says, "you'd be thinking of your input space ... as the xy plane".
Diagram
Observation
A coordinate plane with axes labeled $x$ and $y$ is drawn below the earlier formulas.

Symbol

xy\text{-plane}

Meaning

The two-dimensional input space for a function of two variables; each point in the plane is an input.

Domain

Input space for $f(x,y)$ .

$f(x,y)$

Clear evidence

Shown in the video

Evidence

Formula
Observation
Top-left board shows $f(x,y)=x^2y+\sin(y)$ .
Audio
Observation
The speaker discusses this multivariable function while explaining partial derivatives.

Symbol

$f(x,y)$

Meaning

A two-variable real-valued function used as the example for partial derivatives.

Domain

Defined on variables $x$ and $y$ ; the worked evaluation is at $(1,2)$ .

Knowledge points · 16

Example of a scalar-valued multivariable function

Clear evidence

Shown in the video

Evidence

Audio
Observation
The speaker introduces "some multivariable function like f of x y" and says it has "a two-variable input".
Formula
Observation
The board writes $f(x,y)=x^2y+\sin(y)$ .
Audio
Observation
The speaker adds, "So it'll output just a single number. It's a scalar valued function."

Definition

Explanation

The clip begins by defining a concrete two-variable function whose output is a single number. The speaker explicitly identifies it as scalar-valued.

Formula

f(x,y)=x^2y+\sin(y)

Conditions

The function has two input variables, $x$ and $y$ .
The output is a single number.

What a partial derivative is meant to answer

Clear evidence

Shown in the video

Evidence

Audio
Observation
The speaker asks, "Question is, how do we take the derivative of an expression like this?"
Audio
Observation
The speaker answers, "And there's a certain method called a partial derivative, which is very similar to ordinary derivatives".

Definition

Explanation

A partial derivative is introduced as the method for differentiating a multivariable expression. The video frames it as closely related to ordinary derivatives rather than as a completely separate operation.

Formula

Conditions

Applies when differentiating a function with multiple input variables.

Prerequisites

Example of a scalar-valued multivariable function

Leibniz notation for an ordinary derivative

Clear evidence

Shown in the video

Evidence

Audio
Observation
The speaker says, "if you have something like f of x is equal to x squared" and "I'll use the Leibniz notation here, df dx".
Formula
Observation
The board shows $f(x)=x^2$ and $\frac{df}{dx}(2)$ .

Formula

Explanation

Before moving to partial derivatives, the video reviews the notation $\frac{df}{dx}$ for the derivative of a single-variable function and evaluates it at a point.

Formula

\frac{df}{dx}(2)

Conditions

Used here for the single-variable function $f(x)=x^2$ .
The example evaluates the derivative at $x=2$ .

Prerequisites

Example of a scalar-valued multivariable function

Graphical interpretation of $\frac{df}{dx}$ as slope

Clear evidence

Shown in the video

Evidence

Audio
Observation
The speaker says, "I really like this notation because it's suggestive of what's going on if we sketch out a graph".
Diagram
Observation
A graph of $f(x)=x^2$ is drawn with axes labeled by input and output.
Audio
Observation
The speaker explains, "This little dx here ... a little nudge in the x direction" and "df ... is the resulting change in the output".
Audio
Observation
The speaker concludes, "when you're thinking in terms of graphs, this is slope" and describes it as "rise over run".

Method

Explanation

The video interprets $dx$ as a tiny horizontal input change and $df$ as the resulting vertical output change. Their ratio is presented as the slope of the graph at the chosen point.

Formula

\frac{df}{dx}

Conditions

The function is being viewed as a graph with input on one axis and output on another.
The interpretation is local, depending on where the input point is chosen.

Prerequisites

Leibniz notation for an ordinary derivative

Derivative as a mapping between input and output number lines

Clear evidence

Shown in the video

Evidence

Audio
Observation
The speaker says, "But you could also think about this without graphs if you really wanted to".
Audio
Observation
The speaker describes "your input space is just a number line, and your output space also is just a number line".
Diagram
Observation
Two parallel number lines are drawn, one labeled as input and one as output, with arrows showing a small shift on the input line inducing a larger shift on the output line.
Audio
Observation
The speaker gives the example, "maybe that causes a nudge that's, you know, four times as big, and that would mean your derivative is four at that point".

Method

Explanation

The clip presents a non-graphical interpretation of the ordinary derivative: the function maps points from an input number line to an output number line, and the derivative measures how a small input nudge is amplified into an output nudge.

Formula

Conditions

The function is single-variable.
The interpretation treats input and output spaces as separate number lines.

Prerequisites

Leibniz notation for an ordinary derivative

Interpretation of $\frac{\partial f}{\partial x}$ for a multivariable function

Clear evidence

Shown in the video

Evidence

Audio
Observation
The speaker says, "over in the multivariable world, we can pretty much do the same thing".
Formula
Observation
The board shows $\frac{\partial f}{\partial x}$ .
Audio
Observation
The speaker interprets it as "how does a tiny change in the input in the x direction influence the output?"

Uncertainties

The spoken phrase says "df dx," while the displayed notation is the partial derivative symbol $\frac{\partial f}{\partial x}$ .

Definition

Explanation

The video transfers the ordinary-derivative intuition to the multivariable setting: $\frac{\partial f}{\partial x}$ asks how a tiny change in the input specifically in the $x$ direction affects the output.

Formula

\frac{\partial f}{\partial x}

Conditions

The function has multiple input variables.
The change is taken in the $x$ direction.

Prerequisites

Example of a scalar-valued multivariable function
What a partial derivative is meant to answer
Derivative as a mapping between input and output number lines

For a two-variable function, the input space is the $xy$ -plane

Clear evidence

Shown in the video

Evidence

Audio
Observation
The speaker says, "you'd be thinking of your input space ... as the xy plane".
Audio
Observation
The speaker emphasizes, "this time this is not going to be graphing the function. This is every point on the plane is an input".
Diagram
Observation
A coordinate plane with axes labeled $x$ and $y$ is drawn beneath the formulas.

Definition

Explanation

In the multivariable setting, the drawn plane is not the graph of the function. Instead, each point in the plane represents an input pair to the function.

Formula

Conditions

The function has two input variables.
The plane shown represents input space, not output graph space.

Prerequisites

Example of a scalar-valued multivariable function
Interpretation of $\frac{\partial f}{\partial x}$ for a multivariable function

Partial derivative as directional sensitivity of a multivariable function

Clear evidence

Shown in the video

Evidence

Audio
Observation
The speaker asks how a change in one input influences the output and explains that the function maps points on the plane to a number line.
Diagram
Observation
An input plane with $dx$ and later $dy$ arrows is connected by a curved arrow to an output number line with $df$ arrows.

Definition

Explanation

The video defines a partial derivative informally as the way the output changes when only one input direction is varied. For $f(x,y)$ , changing $x$ gives one output response, and changing $y$ gives another; each measures sensitivity in only one coordinate direction.

Formula

Conditions

The function has more than one input variable.
The discussion isolates change in one coordinate direction at a time.

Notation for partial derivatives

Clear evidence

Shown in the video

Evidence

Audio
Observation
The speaker says people use a new notation with a $d$ that has a curl at the top, often read as “partial.”
Formula
Observation
The written expressions are changed from $\frac{df}{dx}$ and $\frac{df}{dy}$ to $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ .

Definition

Explanation

Partial derivatives are written with the symbol $\partial$ instead of the ordinary differential symbol $d$ . The video presents this notation as a way to signal that a multivariable function is involved and that only one variable’s direction is being considered.

Formula

\frac{\partial f}{\partial x}, \quad \frac{\partial f}{\partial y}

Conditions

Used for derivatives of multivariable functions.
Distinguishes partial derivatives from ordinary single-variable derivatives.

Prerequisites

Partial derivative as directional sensitivity of a multivariable function

Computing a partial derivative by holding the other variable fixed

Clear evidence

Shown in the video

Evidence

Audio
Observation
The speaker says that when evaluating $\frac{\partial f}{\partial x}(1,2)$ , it only cares about movement in the $x$ direction, so it treats $y$ as a constant.
Formula
Observation
The board rewrites the problem as $\frac{\partial}{\partial x}\left(x^2\cdot 2+\sin(2)\right)\big|_{x=1}$ .

Method

Explanation

To compute $\frac{\partial f}{\partial x}$ at a point, the video substitutes the fixed value of the other variable first, turning the expression into an ordinary one-variable derivative in $x$ . In the example, $y=2$ is inserted before differentiating.

Formula

\frac{\partial f}{\partial x}(1,2)=\left.\frac{\partial}{\partial x}\left(x^2\cdot 2+\sin(2)\right)\right|_{x=1}

Conditions

The target is a partial derivative with respect to one variable.
The other independent variable is held fixed at its evaluation value.

Prerequisites

Notation for partial derivatives

Partial derivative at a point

Clear evidence

Shown in the video

Evidence

Formula
Observation
Board explicitly writes $\frac{\partial f}{\partial x}(1,2)$ and $\frac{\partial f}{\partial y}(1,2)$ under the title “Partial derivatives.”
Audio
Observation
Speaker calls these partial derivatives at a point and computes each one by substituting the fixed coordinate before differentiating in the other variable.

Definition

Explanation

The clip treats $\frac{\partial f}{\partial x}(1,2)$ and $\frac{\partial f}{\partial y}(1,2)$ as derivatives of $f$ taken in one coordinate direction while the other coordinate is held fixed at the specified point. For $\frac{\partial f}{\partial x}(1,2)$ , $y$ is fixed at $2$ first; for $\frac{\partial f}{\partial y}(1,2)$ , $x$ is fixed at $1$ first.

Formula

\frac{\partial f}{\partial x}(1,2),\quad \frac{\partial f}{\partial y}(1,2)

Conditions

The function must have an expression in two variables.
One coordinate is fixed while differentiating with respect to the other.
The result is a number, not a formula in $x$ and $y$ .

General partial derivative as a function of $(x,y)$

Clear evidence

Shown in the video

Evidence

Audio
Observation
Speaker says that often you do not just compute at a point but want a general formula that tells you what happens if you plug in any point $(x,y)$ .
Formula
Observation
New line writes $\frac{\partial f}{\partial x}(x,y)=\frac{\partial}{\partial x}(x^2y+\sin(y))$ and simplifies to $2xy+0$ .

Definition

Explanation

Instead of substituting numerical constants ahead of time, the speaker rewrites the same procedure symbolically: treat the non-differentiated variable as a constant and differentiate with respect to the chosen variable. The result is a formula valid at any input point.

Formula

\frac{\partial f}{\partial x}(x,y)=\frac{\partial}{\partial x}\bigl(x^2y+\sin(y)\bigr)

Conditions

Use when the desired output is a rule depending on $(x,y)$ rather than a single numerical value.
The other variable is treated as constant during differentiation.

Prerequisites

Partial derivative at a point

Claims and conditions · 5

A partial derivative with respect to one variable ignores changes in the other variable

Clear evidence

Shown in the video

Evidence

Audio
Observation
The speaker states that $\frac{\partial f}{\partial x}$ only cares about movement in the $x$ direction and treats $y$ as a constant.

Proposition

Statement

For a function $f(x,y)$ , the partial derivative with respect to $x$ measures only the effect of changing $x$ ; the other variable is treated as fixed.

Hypotheses

$f$ is a function of multiple variables.
The derivative being discussed is with respect to one chosen variable.

Quantifiers

For the worked example $f(x,y)$ at $(1,2)$ , when computing $\frac{\partial f}{\partial x}$ , $y$ is held constant.

After freezing the other variable, the partial derivative becomes an ordinary derivative

Clear evidence

Shown in the video

Evidence

Audio
Observation
The speaker says, “And here, this is actually just an ordinary derivative.”
Formula
Observation
After substituting $y=2$ , the displayed expression becomes a derivative in $x$ only.

Proposition

Statement

Once the non-target variable is substituted by a constant, the remaining computation is an ordinary single-variable derivative.

Hypotheses

The original expression is a partial derivative of a multivariable function.
All variables except the differentiation variable have been fixed to constants.

Quantifiers

In the example, after replacing $y$ by $2$ , the expression depends only on $x$ .

Value of the $x$ -partial at $(1,2)$

Clear evidence

Shown in the video

Evidence

Formula
Observation
Board shows $\frac{\partial f}{\partial x}(1,2)=\frac{\partial}{\partial x}(x^2\cdot 2+\sin(2))\big|_{x=1}=4x+0\big|_{x=1}=4$ .
Audio
Observation
Speaker states the derivative of $x^2\cdot 2$ is $4x$ , the derivative of $\sin(2)$ is $0$ , and evaluating at $x=1$ gives $4$ .

Proposition

Statement

For $f(x,y)=x^2y+\sin(y)$ , one has $\frac{\partial f}{\partial x}(1,2)=4$ .

Hypotheses

$f(x,y)=x^2y+\sin(y)$
Evaluate with respect to $x$ while holding $y=2$ fixed
Then substitute $x=1$

Quantifiers

At the specific point $(1,2)$ .

Value of the $y$ -partial at $(1,2)$

Clear evidence

Shown in the video

Evidence

Formula
Observation
Board shows $\frac{\partial f}{\partial y}(1,2)=\frac{\partial}{\partial y}((1)^2y+\sin(y))\big|_{y=2}=1+\cos(y)\big|_{y=2}=1+\cos(2)$ .
Audio
Observation
Speaker says the derivative of $1\cdot y$ is $1$ , the derivative of $\sin(y)$ is $\cos(y)$ , and evaluating at $y=2$ gives $1+\cos(2)$ .

Proposition

Statement

For $f(x,y)=x^2y+\sin(y)$ , one has $\frac{\partial f}{\partial y}(1,2)=1+\cos(2)$ .

Hypotheses

$f(x,y)=x^2y+\sin(y)$
Evaluate with respect to $y$ while holding $x=1$ fixed
Then substitute $y=2$

Quantifiers

At the specific point $(1,2)$ .

General formula for the $x$ -partial

Clear evidence

Shown in the video

Evidence

Formula
Observation
Board shows $\frac{\partial f}{\partial x}(x,y)=\frac{\partial}{\partial x}(x^2y+\sin(y))=2xy+0$ .
Audio
Observation
Speaker says the derivative of $x^2$ times a constant is $2x$ times that constant, and the derivative of a constant is zero.

Uncertainties

The final simplified form is left as $2xy+0$ on screen rather than rewritten as $2xy$ .

Proposition

Statement

For $f(x,y)=x^2y+\sin(y)$ , the general partial derivative with respect to $x$ is $\frac{\partial f}{\partial x}(x,y)=2xy+0$ , i.e. $2xy$ .

Hypotheses

$f(x,y)=x^2y+\sin(y)$
Treat $y$ as constant while differentiating with respect to $x$

Quantifiers

For arbitrary $(x,y)$ .

Derivations and proofs · 8

Two equivalent ways to understand the ordinary derivative before introducing partial derivatives

Clear evidence

Shown in the video

Evidence

Formula
Observation
The board shows $f(x)=x^2$ and $\frac{df}{dx}(2)$ .
Diagram
Observation
A parabola is drawn with a marked point at $x=2$ , a horizontal arrow $dx$ , and a vertical arrow $df$ .
Diagram
Observation
Two number lines are drawn to represent input and output spaces, with arrows showing a small input shift producing a larger output shift.
Audio
Observation
The speaker first explains the graph/slope interpretation and then says, "But you could also think about this without graphs if you really wanted to".

Intuitive argument

Steps

Expression
$f(x)=x^2$
Explanation
Start with a single-variable example function.
Justification
The speaker explicitly writes and names this example.
Shown in the video
Expression
$\frac{df}{dx}(2)$
Explanation
Introduce Leibniz notation for the derivative evaluated at $x=2$ .
Justification
The speaker says he will use Leibniz notation and evaluate at 2.
Shown in the video
Expression
$dx \text{ as a horizontal nudge},\quad df \text{ as the resulting vertical change}$
Explanation
On the graph, interpret $dx$ as a tiny input change and $df$ as the induced output change.
Justification
The speaker verbally defines both symbols while drawing arrows on the parabola.
Shown in the video
Expression
$\frac{df}{dx}=\text{slope}$
Explanation
Conclude that the derivative is the local rise-over-run slope at the chosen point.
Justification
The speaker states, "when you're thinking in terms of graphs, this is slope."
Shown in the video
Expression
$\text{input number line} \to \text{output number line}$
Explanation
Reinterpret the same derivative without a graph by viewing the function as a map between two number lines.
Justification
The speaker explicitly offers this alternative interpretation.
Shown in the video
Expression
$\text{a small input nudge can become an output nudge four times as large}$
Explanation
Use the example to say the derivative at that point is 4.
Justification
The speaker gives this numerical illustration in words.
Shown in the video

Conclusion

The ordinary derivative can be understood either as local slope on a graph or as amplification of a tiny input change under a number-line mapping; this prepares the same style of reasoning for partial derivatives.

Transferring the derivative intuition from one variable to two variables

Clear evidence

Shown in the video

Evidence

Audio
Observation
The speaker says, "over in the multivariable world, we can pretty much do the same thing".
Formula
Observation
The board shows $\frac{\partial f}{\partial x}$ and later $(1,2)$ .
Diagram
Observation
An $xy$ -plane is drawn and a point corresponding to $(1,2)$ is marked.
Audio
Observation
The speaker says the input space is the $xy$ plane and that every point on the plane is an input.

Uncertainties

The final sentence is cut off after "this tiny change dx"; no completed visual explanation of the multivariable increment is shown within the clip.

Intuitive argument

Steps

Expression
$\text{Use the same idea in the multivariable case}$
Explanation
After explaining ordinary derivatives, the speaker announces that the multivariable case works similarly.
Justification
Directly stated in the audio.
Shown in the video
Expression
$\frac{\partial f}{\partial x}$
Explanation
Write the partial-derivative notation for changes in the $x$ direction.
Justification
The symbol appears on the board and is verbally interpreted.
Shown in the video
Expression
$\text{How does a tiny change in the input in the }x\text{ direction influence the output?}$
Explanation
State the meaning of the partial derivative as a directional input-output sensitivity question.
Justification
The speaker gives this interpretation in words.
Shown in the video
Expression
$\text{input space} = xy\text{-plane}$
Explanation
Replace the one-dimensional input number line with a two-dimensional input plane.
Justification
The speaker explicitly says the input space is the $xy$ plane.
Shown in the video
Expression
$(1,2)$
Explanation
Choose a specific input point in the plane for the example.
Justification
The speaker says, "let's say you were evaluating this at a point like 1, 2," and marks it on the plane.
Shown in the video
Expression
$\text{tiny change }dx\text{ at }(1,2)$
Explanation
Begin to describe a small change in the input at the chosen point.
Justification
The speaker starts this explanation, but the clip ends before the full visual argument is completed.
Shown in the video

Conclusion

For a function of two variables, $\frac{\partial f}{\partial x}$ is introduced as the analogue of the ordinary derivative: it measures how the output responds to a tiny input change in the $x$ direction, now visualized on the $xy$ input plane rather than on a single number line.

Rewriting $\frac{\partial f}{\partial x}(1,2)$ as a one-variable derivative problem

Clear evidence

Shown in the video

Evidence

Formula
Observation
The board writes $\frac{\partial f}{\partial x}(1,2)=\frac{\partial}{\partial x}\left(x^2\cdot 2+\sin(2)\right)\big|_{x=1}$ .
Audio
Observation
The speaker explains that because only $x$ matters, $y$ can be plugged in ahead of time.

Uncertainties

The final numerical evaluation is not reached within the clip.

Proof

Steps

Expression
$f(x,y)=x^2y+\sin(y)$
Explanation
Start from the given multivariable function.
Justification
Given on the board.
Shown in the video
Expression
$\frac{\partial f}{\partial x}(1,2)$
Explanation
Identify the quantity to evaluate: the partial derivative with respect to $x$ at the point $(1,2)$ .
Justification
Written explicitly on the board.
Shown in the video
Expression
$y=2 \text{ is held fixed}$
Explanation
Because the derivative is with respect to $x$ , the $y$ -coordinate is treated as constant.
Justification
Stated verbally by the speaker.
Shown in the video
Expression
$\frac{\partial}{\partial x}\left(x^2\cdot 2+\sin(2)\right)\big|_{x=1}$
Explanation
Substitute $y=2$ into the formula before differentiating, leaving an expression depending only on $x$ , then indicate evaluation at $x=1$ .
Justification
Shown as the rewritten board expression and justified by the speaker’s explanation.
Shown in the video

Conclusion

The partial derivative problem has been reduced to differentiating a single-variable expression and then evaluating at $x=1$ ; the clip stops before carrying out that final derivative and substitution.

Derivation of $\frac{\partial f}{\partial x}(1,2)$

Clear evidence

Shown in the video

Evidence

Formula
Observation
Step-by-step algebra is written on the upper-right board.
Audio
Observation
Speaker narrates each differentiation and substitution step.

Proof

Steps

Expression
$\frac{\partial f}{\partial x}(1,2)=\frac{\partial}{\partial x}\bigl(x^2\cdot 2+\sin(2)\bigr)\Big|_{x=1}$
Explanation
Start from the given function and freeze $y$ at $2$ before differentiating with respect to $x$ .
Justification
Definition of a partial derivative at a point, as demonstrated by the speaker.
Shown in the video
Expression
$=4x+0\Big|_{x=1}$
Explanation
Differentiate $x^2\cdot 2$ to get $4x$ , and differentiate the constant $\sin(2)$ to get $0$ .
Justification
Power rule for $x^2$ and the fact that the derivative of a constant is zero.
Shown in the video
Expression
$=4$
Explanation
Substitute $x=1$ into $4x+0$ .
Justification
Evaluation at the specified point.
Shown in the video

Conclusion

$\frac{\partial f}{\partial x}(1,2)=4$ .

Derivation of $\frac{\partial f}{\partial y}(1,2)$

Clear evidence

Shown in the video

Evidence

Formula
Observation
Red handwriting builds the full computation line by line.
Audio
Observation
Speaker explains that $x$ stays constant at $1$ , then differentiates with respect to $y$ and evaluates at $y=2$ .

Proof

Steps

Expression
$\frac{\partial f}{\partial y}(1,2)=\frac{\partial}{\partial y}\bigl((1)^2y+\sin(y)\bigr)\Big|_{y=2}$
Explanation
Freeze $x$ at $1$ and rewrite the function as a one-variable expression in $y$ before differentiating.
Justification
Definition of a partial derivative at a point, with the other coordinate held fixed.
Shown in the video
Expression
$=1+\cos(y)\Big|_{y=2}$
Explanation
Differentiate $(1)^2y=y$ to get $1$ , and differentiate $\sin(y)$ to get $\cos(y)$ .
Justification
Basic derivative rules for a linear term and for sine.
Shown in the video
Expression
$=1+\cos(2)$
Explanation
Substitute $y=2$ into the differentiated expression.
Justification
Evaluation at the specified point.
Shown in the video

Conclusion

$\frac{\partial f}{\partial y}(1,2)=1+\cos(2)$ .

Derivation of the general formula for $\frac{\partial f}{\partial x}(x,y)$

Clear evidence

Shown in the video

Evidence

Formula
Observation
After clearing space, the board writes $\frac{\partial f}{\partial x}(x,y)=\frac{\partial}{\partial x}(x^2y+\sin(y))=2xy+0$ .
Audio
Observation
Speaker says this is the same idea, except now we pretend the other variable is constant instead of plugging in a number ahead of time.

Uncertainties

The clip ends before the speaker explicitly simplifies $2xy+0$ to $2xy$ on screen.

Proof

Steps

Expression
$\frac{\partial f}{\partial x}(x,y)=\frac{\partial}{\partial x}\bigl(x^2y+\sin(y)\bigr)$
Explanation
Rewrite the task as finding a formula valid for arbitrary $(x,y)$ rather than a single point value.
Justification
Transition from point evaluation to general partial derivative, as stated by the speaker.
Shown in the video
Expression
$=2xy+0$
Explanation
Treat $y$ as a constant multiplier in $x^2y$ , giving $2x\cdot y=2xy$ ; treat $\sin(y)$ as a constant term, giving $0$ .
Justification
Constant-multiple rule, power rule, and derivative of a constant.
Shown in the video

Conclusion

$\frac{\partial f}{\partial x}(x,y)=2xy+0$ , equivalently $2xy$ .

Calculation of ∂ $f/y$

Clear evidence

Shown in the video

Evidence

Audio
Observation
We write down all of the same things, now you're taking it with respect to y... But this time we're considering all of the x's to be constants... when you take the derivative with respect to y of some kind of constant times y, it's just going to equal that constant... over here, you're taking the derivative of sine of y, there's no x's in there, so that remains the cosine of y.
Formula
Observation
∂f/∂ $y (x,y)$ = ∂/∂ $y (x^2y + \sin (y)) = x^2 + \cos (y)$

Proof

Steps

Expression
$f(x,y) = x^2y + sin(y)$
Explanation
Start with the given function.
Justification
Given.
Shown in the video
Expression
$∂f/∂y (x,y) = ∂/∂y (x^2y + sin(y))$
Explanation
Set up the partial derivative with respect to y.
Justification
Definition of partial derivative.
Shown in the video
Expression
$= ∂/∂y (x^2y) + ∂/∂y (sin(y))$
Explanation
Apply the sum rule for derivatives.
Justification
Sum rule of differentiation.
Derived from the video
Expression
$= x^2 * ∂/∂y (y) + cos(y)$
Explanation
Treat $x^2$ as a constant and pull it out; apply the derivative of $\sin (y)$ .
Justification
Constant multiple rule and standard trigonometric derivative.
Shown in the video
Expression
$= x^2 * 1 + cos(y)$
Explanation
The derivative of y with respect to y is 1.
Justification
Power rule for derivative.
Shown in the video
Expression
$= x^2 + cos(y)$
Explanation
Simplify the expression.
Justification
Algebraic simplification.
Shown in the video

Conclusion

The partial derivative of $f(x,y) = x^2y + \sin (y)$ with respect to y is $x^2 + \cos (y)$ .

Calculation of ∂f/∂x

Clear evidence

Shown in the video

Evidence

Formula
Observation
∂f/∂ $x (x,y)$ = ∂/∂ $x (x^2y + \sin (y)) = 2$ xy + 0

Proof

Steps

Expression
$f(x,y) = x^2y + sin(y)$
Explanation
Start with the given function.
Justification
Given.
Shown in the video
Expression
$∂f/∂x (x,y) = ∂/∂x (x^2y + sin(y))$
Explanation
Set up the partial derivative with respect to x.
Justification
Definition of partial derivative.
Shown in the video
Expression
$= ∂/∂x (x^2y) + ∂/∂x (sin(y))$
Explanation
Apply the sum rule for derivatives.
Justification
Sum rule of differentiation.
Derived from the video
Expression
$= y * ∂/∂x (x^2) + 0$
Explanation
Treat y as a constant and pull it out; the derivative of $\sin (y)$ with respect to x is 0 since $\sin (y)$ is constant with respect to x.
Justification
Constant multiple rule and derivative of a constant.
Shown in the video
Expression
$= y * 2x + 0$
Explanation
Apply the power rule to $x^2$ .
Justification
Power rule for derivative.
Shown in the video
Expression
$= 2xy$
Explanation
Simplify the expression.
Justification
Algebraic simplification.
Shown in the video

Conclusion

The partial derivative of $f(x,y) = x^2y + \sin (y)$ with respect to x is 2xy.

Worked examples · 5

Ordinary derivative example: $f(x)=x^2$ at $x=2$

Clear evidence

Shown in the video

Evidence

Formula
Observation
The board shows $f(x)=x^2$ and $\frac{df}{dx}(2)$ .
Diagram
Observation
A parabola is drawn with a marked point at $x=2$ and arrows labeled $dx$ and $df$ .
Audio
Observation
The speaker explains the graph as slope and then gives a number-line interpretation where the output nudge is four times as big.

Uncertainties

The value 4 is stated verbally as an illustrative outcome of the mapping interpretation; the clip does not show a formal algebraic computation of $f'(2)$ .

Problem

Explain what $\frac{df}{dx}(2)$ means for the function $f(x)=x^2$ .

Given

$f(x)=x^2$
Evaluate at $x=2$
Use Leibniz notation $\frac{df}{dx}$

Goal

Interpret the derivative geometrically and as an input-output mapping.

Steps

Expression
$\text{Draw the graph of }f(x)=x^2$
Explanation
Represent the function as a parabola with input and output axes.
Justification
The speaker sketches the graph to explain the notation.
Shown in the video
Expression
$\text{Mark }x=2\text{ and add }dx,df$
Explanation
Show a tiny horizontal input change and the resulting vertical output change at that point.
Justification
The speaker labels these increments on the graph.
Shown in the video
Expression
$\frac{df}{dx}=\text{slope}$
Explanation
Identify the derivative with the local rise-over-run slope at $x=2$ .
Justification
Stated directly by the speaker.
Shown in the video
Expression
$\text{Alternatively, map one number line to another}$
Explanation
View the function as sending input values to output values without using a graph.
Justification
The speaker explicitly offers this non-graphical interpretation.
Shown in the video
Expression
$\text{If the output nudge is four times the input nudge, the derivative is }4$
Explanation
Use the size ratio of the induced output change to the input change as the derivative value in the example.
Justification
The speaker states this numerical illustration verbally.
Shown in the video

Answer

The derivative at $x=2$ is interpreted as the local slope of $f(x)=x^2$ , and in the mapping picture as an amplification factor; the speaker illustrates this with the value 4.

Verification

The interpretation is checked visually by the graph arrows and verbally by the speaker's statement that the ratio of output change to input change is the slope.

Partial derivative setup for a two-variable function at $(1,2)$

Clear evidence

Shown in the video

Evidence

Formula
Observation
The board shows $\frac{\partial f}{\partial x}$ and $(1,2)$ .
Diagram
Observation
An $xy$ -plane is drawn and a point corresponding to $(1,2)$ is marked.
Audio
Observation
The speaker says the input space is the $xy$ plane and that every point on the plane is an input.
Audio
Observation
The speaker begins, "And then you'd say, okay, so this tiny nudge in the input, this tiny change dx" before the clip ends.

Uncertainties

The example is only set up; the clip ends before the speaker completes the explanation of the tiny change at $(1,2)$ .
No numerical value for the partial derivative is given in this segment.

Problem

Introduce how to think about $\frac{\partial f}{\partial x}$ for a function of two variables at the input point $(1,2)$ .

Given

A multivariable function $f(x,y)$ has been introduced earlier.
The notation $\frac{\partial f}{\partial x}$ is written.
The input point $(1,2)$ is chosen.
The input space is represented by the $xy$ -plane.

Goal

Set up the geometric interpretation of a partial derivative in the multivariable case.

Steps

Expression
$\text{Switch from one input number line to the }xy\text{-plane}$
Explanation
Replace the single-variable input space with a two-dimensional input space.
Justification
The speaker explicitly says the input space is now the $xy$ plane.
Shown in the video
Expression
$\text{Every point on the plane is an input}$
Explanation
Clarify that the plane is not the graph of the function but the domain of inputs.
Justification
Stated directly by the speaker.
Shown in the video
Expression
$(1,2)$
Explanation
Choose a specific input point at which to consider the partial derivative.
Justification
The speaker says, "let's say you were evaluating this at a point like 1, 2," and marks it on the plane.
Shown in the video
Expression
$\text{Begin describing a tiny change }dx\text{ at that point}$
Explanation
Start the same input-nudge reasoning used for ordinary derivatives, now in the multivariable setting.
Justification
The speaker begins this explanation, but the clip cuts off before completion.
Shown in the video

Answer

The clip establishes that $\frac{\partial f}{\partial x}$ should be thought of as the effect of a tiny input change in the $x$ direction at a point such as $(1,2)$ in the $xy$ input plane.

Verification

Verification is only partial within this segment: the setup is visually shown on the plane, but the full explanatory step is not completed before the clip ends.

Setting up $\frac{\partial f}{\partial x}(1,2)$ for $f(x,y)=x^2y+\sin(y)$

Clear evidence

Shown in the video

Evidence

Formula
Observation
The worked setup is written as $\frac{\partial f}{\partial x}(1,2)=\frac{\partial}{\partial x}\left(x^2\cdot 2+\sin(2)\right)\big|_{x=1}$ .
Audio
Observation
The speaker narrates the substitution of $y=2$ and says the remaining task is an ordinary derivative.

Uncertainties

The derivative is not completed inside the clip.
No final numeric answer is shown or spoken.

Problem

Evaluate the partial derivative of $f(x,y)=x^2y+\sin(y)$ with respect to $x$ at the point $(1,2)$ .

Given

$f(x,y)=x^2y+\sin(y)$
Evaluation point $(1,2)$
Differentiate with respect to $x$

Goal

Reduce the partial derivative to a one-variable derivative and prepare evaluation at $x=1$ .

Steps

Expression
$\text{Treat } y \text{ as constant because we differentiate with respect to } x.$
Explanation
The speaker explains that $\frac{\partial f}{\partial x}$ only cares about movement in the $x$ direction.
Justification
Verbal explanation in the clip.
Shown in the video
Expression
$\text{Substitute } y=2 \text{ into } f(x,y).$
Explanation
Replace every occurrence of $y$ by the fixed value $2$ .
Justification
Explicitly stated and shown on the board.
Shown in the video
Expression
$\frac{\partial}{\partial x}\left(x^2\cdot 2+\sin(2)\right)\big|_{x=1}$
Explanation
This yields a single-variable expression in $x$ , with evaluation at $x=1$ indicated afterward.
Justification
Displayed formula on the board.
Shown in the video

Answer

The clip ends after setting up $\left.\frac{\partial}{\partial x}\left(x^2\cdot 2+\sin(2)\right)\right|_{x=1}$ ; it does not state the final value.

Verification

No verification is performed within the clip.

Worked example: partial derivatives of $f(x,y)=x^2y+\sin(y)$

Clear evidence

Shown in the video

Evidence

Formula
Observation
Entire clip works from $f(x,y)=x^2y+\sin(y)$ to numerical point values and then to a general formula.
Audio
Observation
Speaker explicitly frames the computations as practice and then as a more general formula.

Problem

Given $f(x,y)=x^2y+\sin(y)$ , compute $\frac{\partial f}{\partial x}(1,2)$ and $\frac{\partial f}{\partial y}(1,2)$ , then find a general formula for $\frac{\partial f}{\partial x}(x,y)$ .

Given

$f(x,y)=x^2y+\sin(y)$
Evaluation point $(1,2)$

Goal

Find the two partial derivatives at $(1,2)$ .；Find the general $x$ -partial formula.

Steps

Expression
$\frac{\partial f}{\partial x}(1,2)=\frac{\partial}{\partial x}(x^2\cdot 2+\sin(2))\Big|_{x=1}=4x+0\Big|_{x=1}=4$
Explanation
Hold $y=2$ fixed, differentiate with respect to $x$ , then substitute $x=1$ .
Justification
Partial derivative at a point plus basic differentiation rules.
Shown in the video
Expression
$\frac{\partial f}{\partial y}(1,2)=\frac{\partial}{\partial y}((1)^2y+\sin(y))\Big|_{y=2}=1+\cos(y)\Big|_{y=2}=1+\cos(2)$
Explanation
Hold $x=1$ fixed, differentiate with respect to $y$ , then substitute $y=2$ .
Justification
Partial derivative at a point plus basic differentiation rules.
Shown in the video
Expression
$\frac{\partial f}{\partial x}(x,y)=\frac{\partial}{\partial x}(x^2y+\sin(y))=2xy+0$
Explanation
Repeat the same logic symbolically, treating $y$ as constant rather than substituting a number first.
Justification
Generalization from pointwise computation to a formula in $(x,y)$ .
Shown in the video

Answer

$\frac{\partial f}{\partial x}(1,2)=4$ , $\frac{\partial f}{\partial y}(1,2)=1+\cos(2)$ , and $\frac{\partial f}{\partial x}(x,y)=2xy+0$ (i.e. $2xy$ ).

Verification

Substituting $(x,y)=(1,2)$ into the general formula $2xy$ gives $2\cdot 1\cdot 2=4$ , matching the earlier pointwise result for $\frac{\partial f}{\partial x}(1,2)$ .

Evaluating Partial Derivatives at a Point

Clear evidence

Shown in the video

Evidence

Audio
Observation
If you plugged in 1, 2, you would get 1 plus the cosine of 1, which is what we had before.
Formula
Observation
∂f/∂ $x (1,2)$ ; ∂f/∂ $y (1,2)$

Problem

Evaluate the partial derivatives of $f(x,y) = x^2y + \sin (y)$ at the point (1, 2).

Given

$f(x,y) = x^2y + \sin (y)$
Point (x, y) = (1, 2)

Goal

Find the numerical values of ∂f/∂ $x(1,2)$ and ∂f/∂ $y(1,2)$ .

Steps

Expression
$∂f/∂x (x,y) = 2xy$
Explanation
Use the previously calculated partial derivative with respect to x.
Justification
Result from derivation der-calc-partial-x.
Shown in the video
Expression
$∂f/∂x (1,2) = 2(1)(2) = 4$
Explanation
Substitute $x=1$ and $y=2$ into the expression for ∂ $f/x$ .
Justification
Evaluation of a function at a point.
Derived from the video
Expression
$∂f/∂y (x,y) = x^2 + cos(y)$
Explanation
Use the previously calculated partial derivative with respect to y.
Justification
Result from derivation der-calc-partial-y.
Shown in the video
Expression
$∂f/∂y (1,2) = 1^2 + cos(2) = 1 + cos(2)$
Explanation
Substitute $x=1$ and $y=2$ into the expression for ∂f/∂y.
Justification
Evaluation of a function at a point.
Derived from the video

Answer

∂f/∂ $x(1,2) = 4$ and ∂f/∂ $y(1,2) = 1 + \cos (2)$ .

Verification

The speaker mentions that plugging in (1,2) gives $1 + \cos (1)$ , but the board shows (1,2). Based on the formula $x^2+\cos (y)$ , substituting (1,2) yields $1^2+\cos (2)=1+\cos (2)$ . The audio likely contains a slip of the tongue saying 'cosine of 1' instead of 'cosine of 2'. The visual evidence on the board is consistent with the calculation.

Visual events · 13

Graphical depiction of ordinary derivative as local slope

Clear evidence

Shown in the video

Evidence

Diagram
Observation
A coordinate system is drawn and a parabola for $f(x)=x^2$ is sketched.
Diagram
Observation
The point $x=2$ is marked on the input axis.
Diagram
Observation
A horizontal arrow labeled $dx$ and a vertical arrow labeled $df$ are added near the point on the curve.
Audio
Observation
The speaker explains these arrows as a tiny input nudge and the resulting output change, then identifies the ratio as slope.

Objects

Cartesian axes
Parabola for $f(x)=x^2$
Point at $x=2$
Horizontal arrow $dx$
Vertical arrow $df$

Changes

The graph is drawn first.
The evaluation point $x=2$ is marked.
A small horizontal increment $dx$ is added.
A corresponding vertical increment $df$ is added.
The pair of increments is interpreted as rise over run.

Invariants

The underlying function remains $f(x)=x^2$ .
The chosen evaluation point remains $x=2$ during this explanation.

Interpretation

The animation/drawing shows that the derivative at a point is the local ratio of a tiny output change to a tiny input change, i.e. the slope of the graph there.

Number-line mapping interpretation of the derivative

Clear evidence

Shown in the video

Evidence

Diagram
Observation
Two parallel number lines are drawn, one for input and one for output.
Diagram
Observation
Arrows indicate a small shift on the input line and a larger shift on the output line.
Audio
Observation
The speaker says this is a way to think about the derivative without graphs and gives the example of an output nudge four times as big.

Objects

Input number line
Output number line
Small input arrow
Larger output arrow

Changes

The graph-based picture is replaced by two separate lines.
A small movement on the input line is shown.
A larger corresponding movement on the output line is shown.

Invariants

The function is still being treated as a map from inputs to outputs.
The derivative is still interpreted as a ratio of output change to input change.

Interpretation

This visual reframes differentiation as amplification under a mapping: a tiny input displacement produces a proportionally larger or smaller output displacement.

Input-space visualization for a two-variable function

Clear evidence

Shown in the video

Evidence

Diagram
Observation
An $xy$ -plane is drawn beneath the earlier formulas.
Audio
Observation
The speaker says the input space is the $xy$ plane and that every point on the plane is an input.
Formula
Observation
The point $(1,2)$ is written next to $\frac{\partial f}{\partial x}$ .
Diagram
Observation
A point corresponding to $(1,2)$ is marked on the plane.

Uncertainties

The clip ends before the speaker finishes describing the tiny change at the marked point.

Objects

$x$ -axis
$y$ -axis
$xy$ -plane
Marked input point $(1,2)$
Notation $\frac{\partial f}{\partial x}$

Changes

A new coordinate plane is introduced below the earlier work.
The plane is identified as input space rather than graph space.
The specific input point $(1,2)$ is marked.

Invariants

The plane represents inputs to the multivariable function, not outputs plotted against inputs.
The discussion remains focused on changes in the $x$ direction.

Interpretation

The drawing shifts the setting from one-dimensional input space to two-dimensional input space, preparing the geometric meaning of a partial derivative at a point in the domain.

Visualizing a multivariable function as a map from the plane to a number line

Clear evidence

Shown in the video

Evidence

Diagram
Observation
A green input plane with axes $x$ and $y$ is drawn, a yellow horizontal arrow labeled $dx$ appears, and a curved arrow points to a separate output number line labeled $f$ .
Audio
Observation
The speaker says the function maps points on the plane to the number line and asks how much $dx$ changes the output.

Objects

Input plane with axes $x$ and $y$
Yellow arrow labeled $dx$
Curved mapping arrow
Output number line labeled $f$
Arrow labeled $df$ on the output line

Changes

A horizontal perturbation $dx$ is introduced in the input plane.
The corresponding output perturbation $df$ is drawn on the number line.

Invariants

The output remains a single number line rather than a plane.
The example function $f(x,y)=x^2y+\sin(y)$ stays fixed on the board.

Interpretation

The drawing represents how changing only the $x$ -coordinate of an input affects the scalar output.

Adding the $y$ -direction perturbation to the same mapping picture

Clear evidence

Shown in the video

Evidence

Diagram
Observation
A red vertical arrow labeled $dy$ is added on the input plane, and a different output displacement is marked on the number line.
Audio
Observation
The speaker says one can do the same thing with the $y$ variable and that $dy$ is a change in the $y$ direction.

Objects

Input plane
Red vertical arrow labeled $dy$
Output number line
Red output displacement labeled $df$

Changes

A second input direction, vertical in the plane, is introduced.
The output line now shows a different induced displacement associated with $dy$ .

Invariants

The same input plane and output number line remain in use.
The function $f(x,y)$ is unchanged.

Interpretation

The visual contrast shows that perturbing $y$ produces its own separate output response, distinct from perturbing $x$ .

Replacing $df/dx$ and $df/dy$ with partial-derivative notation

Clear evidence

Shown in the video

Evidence

Formula
Observation
The written fractions change from ordinary $d$ notation to $\partial$ notation.
Audio
Observation
The speaker explains that the curled $d$ is used to emphasize a multivariable function.

Objects

Expression $\frac{df}{dx}(1,2)$
Expression $\frac{df}{dy}(1,2)$
Rewritten expressions $\frac{\partial f}{\partial x}(1,2)$ and $\frac{\partial f}{\partial y}(1,2)$

Changes

The numerator and denominator symbols are rewritten using $\partial$ .

Invariants

The evaluated point remains $(1,2)$ .
The underlying function remains $f(x,y)=x^2y+\sin(y)$ .

Interpretation

The board visually marks the transition from informal directional-change language to standard partial-derivative notation.

Erasing the single-variable comparison panel

Clear evidence

Shown in the video

Evidence

Animation
Observation
The right-side one-dimensional analogy panel is selected and erased.
Audio
Observation
The speaker says he will clear the board because the one-dimensional analogy is probably already familiar.

Objects

Right-side panel with $f(x)=x^2$
Graph of $f(x)$
Number-line analogy below the graph

Changes

The entire right-side comparison area disappears.
Space is freed for the worked partial-derivative setup.

Invariants

The left-side multivariable example and diagrams remain visible.

Interpretation

The lesson shifts from analogy back to direct computation for the multivariable example.

Writing the substitution-based setup for the partial derivative

Clear evidence

Shown in the video

Evidence

Formula
Observation
At the top right, the speaker writes $\frac{\partial f}{\partial x}(1,2)=\frac{\partial}{\partial x}\left(x^2\cdot 2+\sin(2)\right)\big|_{x=1}$ .
Audio
Observation
The speaker says they can plug in $y=2$ ahead of time and notes the result is just an ordinary derivative.

Objects

New top-right worked expression
Original function statement on the left
Input/output diagrams below

Changes

A fresh symbolic derivation is written in the cleared space.
The expression explicitly replaces $y$ by $2$ and marks evaluation at $x=1$ .

Invariants

The same function and evaluation point are used throughout.

Interpretation

The visual step converts the abstract partial-derivative question into a concrete one-variable differentiation problem.

Initial whiteboard layout

Clear evidence

Shown in the video

Evidence

Diagram
Observation
The board already contains the title “Partial derivatives,” the function $f(x,y)=x^2y+\sin(y)$ , the two target expressions $\frac{\partial f}{\partial x}(1,2)$ and $\frac{\partial f}{\partial y}(1,2)$ , and a lower sketch with axes and arrows.

Objects

Title “Partial derivatives”
Function definition $f(x,y)=x^2y+\sin(y)$
Two boxed target partial derivatives
Coordinate sketch with $x$ and $y$ axes
Curved arrow to a horizontal line labeled $f$
Arrows labeled $df$ , $dx$ , and $dy$

Changes

The upper-right computation for $\frac{\partial f}{\partial x}(1,2)$ is completed live.
The rest of the board remains visible as reference.

Invariants

The function definition and the two target partial-derivative expressions stay on screen.
The lower directional sketch remains unchanged.

Interpretation

The layout separates the algebraic computation above from a geometric intuition sketch below, linking symbolic partial derivatives to changes in coordinate directions.

Live construction of the $y$ -partial computation

Clear evidence

Shown in the video

Evidence

Animation
Observation
Red handwriting appears sequentially to build the entire $\frac{\partial f}{\partial y}(1,2)$ computation.

Objects

Red expression $\frac{\partial f}{\partial y}(1,2)$
Red substituted form $\frac{\partial}{\partial y}((1)^2y+\sin(y))\big|_{y=2}$
Red simplified result $1+\cos(2)$

Changes

The speaker writes the derivative operator, then the substituted function, then the differentiated expression, then the final evaluated value.

Invariants

The earlier blue/yellow $x$ -partial computation remains above.
The lower sketch remains unchanged.

Interpretation

The color shift to red visually distinguishes the second example and reinforces that the fixed variable has changed from $y=2$ to $x=1$ .

Erasing prior pointwise work to introduce the general formula

Clear evidence

Shown in the video

Evidence

Animation
Observation
A selection box appears around the earlier right-side work, and parts of it are removed to make room.
Audio
Observation
Speaker says, “let me make a little bit of space for ourselves here” and “We don’t need any of this anymore.”

Uncertainties

The exact sequence of deleted versus retained fragments is partially obscured by the editing motion.

Objects

Selection rectangle
Previously written pointwise computations
Remaining header $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ labels

Changes

The detailed numerical work on the right is cleared.
The board is reorganized to leave space for a new symbolic derivation.

Invariants

The function definition and the conceptual sketch remain.
The notation for partial derivatives with respect to $x$ and $y$ is retained.

Interpretation

This visual reset marks the transition from computing numbers at one point to deriving a reusable formula.

Writing the general $x$ -partial formula

Clear evidence

Shown in the video

Evidence

Animation
Observation
New yellow and red writing builds $\frac{\partial f}{\partial x}(x,y)=\frac{\partial}{\partial x}(x^2y+\sin(y))=2xy+0$ .

Objects

Expression $\frac{\partial f}{\partial x}(x,y)$
Substituted symbolic form with $y$ kept general
Simplified result $2xy+0$

Changes

The board shifts from numeric substitution to symbolic treatment of $y$ as a constant.
The final line stops at $2xy+0$ within this clip.

Invariants

The underlying function $f(x,y)=x^2y+\sin(y)$ is unchanged.
The method of treating the other variable as constant is the same as before.

Interpretation

The visual progression shows that the general derivative formula is obtained by the same rule as the pointwise calculation, without first inserting numerical values.

Misconceptions · 7

Partial derivatives are not a wholly different operation from ordinary derivatives

Clear evidence

Shown in the video

Evidence

Audio
Observation
The speaker says a partial derivative "is very similar to ordinary derivatives" and that he wants to show "they're secretly the same thing".

Misconception

One might think partial derivatives are an unrelated or entirely new kind of derivative.

Clarification

The video explicitly frames partial derivatives as very similar to ordinary derivatives and builds them from the same input-change/output-change intuition.

In the multivariable setup, the drawn plane is input space, not the graph of the function

Clear evidence

Shown in the video

Evidence

Audio
Observation
The speaker says, "So this time this is not going to be graphing the function. This is every point on the plane is an input."

Misconception

A viewer may assume the newly drawn $xy$ -plane is another graph of the function itself.

Clarification

The speaker explicitly corrects this by stating that the plane represents the input space, with each point serving as an input pair to the function.

Thinking a partial derivative describes all changes of the function

Clear evidence

Shown in the video

Evidence

Audio
Observation
The speaker says a partial derivative does not tell the full story of how $f$ changes because it only cares about one direction.

Misconception

One might think $\frac{\partial f}{\partial x}$ or $\frac{\partial f}{\partial y}$ gives the complete behavior of $f$ under arbitrary input changes.

Clarification

The video stresses that each partial derivative captures only one coordinate direction; together they are only parts of the full story.

Confusing partial-derivative notation with ordinary derivative notation

Clear evidence

Shown in the video

Evidence

Audio
Observation
The speaker says people use a new notation mostly to emphasize that a multivariable function is involved.
Formula
Observation
The board visibly changes from $d$ to $\partial$ .

Misconception

One might treat $\frac{df}{dx}$ and $\frac{\partial f}{\partial x}$ as interchangeable without recognizing the multivariable context.

Clarification

The video introduces $\partial$ specifically to signal that the derivative is taken with respect to one variable of a multivariable function.

Partial derivative at a point versus partial derivative as a function

Clear evidence

Shown in the video

Evidence

Audio
Observation
Speaker explicitly contrasts computing a partial derivative at a point with wanting a general formula that works for any point $(x,y)$ .

Misconception

Assuming that a partial derivative always means only a single numerical value at one point.

Clarification

The clip distinguishes $\frac{\partial f}{\partial x}(1,2)$ , which is a number, from $\frac{\partial f}{\partial x}(x,y)$ , which is a formula valid for arbitrary inputs.

Forgetting to hold the non-differentiated variable constant

Clear evidence

Shown in the video

Evidence

Audio
Observation
Speaker repeatedly says the other variable is held constant: $x$ stays constant at $1$ for the $y$ -derivative, and later we pretend $y$ is a constant for the general $x$ -derivative.

Misconception

Differentiating all symbols in the expression as though the function had only one variable.

Clarification

When computing $\frac{\partial f}{\partial x}$ , treat $y$ as constant; when computing $\frac{\partial f}{\partial y}$ , treat $x$ as constant.

Misconception: Derivatives are only about graphs and slopes

Clear evidence

Shown in the video

Evidence

Audio
Observation
It's important to understand that graphs and slopes are not the only way to understand derivatives, because as soon as you start thinking about vector-valued functions or functions with inputs of higher dimensions than just two, you can no longer think in terms of graphs...

Misconception

Students might believe that the geometric interpretation of derivatives as slopes of tangent lines on a graph is the only valid way to understand them.

Clarification

While useful for simple cases, the graphical interpretation breaks down for higher-dimensional inputs or vector-valued outputs. A more general understanding involves viewing derivatives as the ratio of an output 'nudge' to an input 'nudge' in a specific direction.

Concept relations · 14

Leibniz notation for an ordinary derivative → Interpretation of $\frac{\partial f}{\partial x}$ for a multivariable function

Clear evidence

Shown in the video

Evidence

Audio
Observation
The speaker first reviews ordinary derivatives and then says, "over in the multivariable world, we can pretty much do the same thing".
Formula
Observation
The board moves from $\frac{df}{dx}$ to $\frac{\partial f}{\partial x}$ .

Prerequisite

Explanation

The clip uses the ordinary derivative and its Leibniz notation as the conceptual foundation for understanding the partial derivative.

Example of a scalar-valued multivariable function → What a partial derivative is meant to answer

Clear evidence

Shown in the video

Evidence

Formula
Observation
The clip begins with $f(x,y)=x^2y+\sin(y)$ and later introduces $\frac{\partial f}{\partial x}$ .
Audio
Observation
The speaker asks how to differentiate such a multivariable expression and answers with partial derivatives.

Application

Explanation

The notion of a partial derivative is introduced specifically to handle differentiation of the multivariable function example.

Graphical interpretation of $\frac{df}{dx}$ as slope → Derivative as a mapping between input and output number lines

Clear evidence

Shown in the video

Evidence

Audio
Observation
The speaker gives the graph interpretation and then says, "But you could also think about this without graphs if you really wanted to".
Diagram
Observation
Both a parabola with $dx,df$ arrows and a pair of number lines are used to explain the same derivative idea.

Equivalent

Explanation

The video presents two different visual interpretations of the ordinary derivative that express the same underlying input-change/output-change relationship.

Derivative as a mapping between input and output number lines → For a two-variable function, the input space is the $xy$ -plane

Clear evidence

Shown in the video

Evidence

Audio
Observation
The speaker first describes input and output as number lines for a single-variable function, then says that in the multivariable case the input space is the $xy$ plane.
Diagram
Observation
The visuals move from two number lines to a two-dimensional input plane.

Generalizes

Explanation

The one-dimensional input-space picture is generalized to a two-dimensional input space when moving from ordinary derivatives to partial derivatives.

Interpretation of $\frac{\partial f}{\partial x}$ for a multivariable function → For a two-variable function, the input space is the $xy$ -plane

Clear evidence

Shown in the video

Evidence

Formula
Observation
The board shows $\frac{\partial f}{\partial x}$ alongside the newly drawn $xy$ -plane and point $(1,2)$ .
Audio
Observation
The speaker interprets the notation as asking how a tiny change in the input in the $x$ direction influences the output, then identifies the input space as the $xy$ plane.

Proof dependency

Explanation

The meaning of $\frac{\partial f}{\partial x}$ in this clip depends on recognizing that the relevant domain is the two-dimensional input plane, not a graph of the function.

Partial derivative as directional sensitivity of a multivariable function → Notation for partial derivatives

Clear evidence

Shown in the video

Evidence

Audio
Observation
The speaker first explains directional changes $dx$ and $dy$ and then introduces the $\partial$ notation.
Formula
Observation
The written expressions are converted from $df/dx$ , $df/dy$ to $\partial f/\partial x$ , $\partial f/\partial y$ .

Prerequisite

Explanation

The informal idea of changing only one input direction is used to motivate the formal partial-derivative notation.

Notation for partial derivatives → Computing a partial derivative by holding the other variable fixed

Clear evidence

Shown in the video

Evidence

Formula
Observation
After introducing $\frac{\partial f}{\partial x}$ , the speaker immediately uses it in the worked setup.

Prerequisite

Explanation

Understanding the $\partial$ notation is necessary before applying the rule that the other variable is held fixed during computation.

Computing a partial derivative by holding the other variable fixed → Setting up $\frac{\partial f}{\partial x}(1,2)$ for $f(x,y)=x^2y+\sin(y)$

Clear evidence

Shown in the video

Evidence

Formula
Observation
The method is instantiated on $f(x,y)=x^2y+\sin(y)$ at $(1,2)$ .

Application

Explanation

The general procedure of freezing the non-target variable is demonstrated directly in the worked example.

Partial derivative as directional sensitivity of a multivariable function → $f(x)=x^2$

Clear evidence

Shown in the video

Evidence

Formula
Observation
The right panel shows $f(x)=x^2$ and $\frac{df}{dx}(2)$ beside the multivariable example.
Audio
Observation
The speaker refers to this as a one-dimensional analogy.

Contrast

Explanation

The multivariable partial-derivative idea is contrasted with the simpler ordinary derivative of a single-variable function.

General partial derivative as a function of $(x,y)$ → Partial derivative at a point

Clear evidence

Shown in the video

Evidence

Audio
Observation
Speaker says the general formula is very similar, except instead of plugging in the constant ahead of time, we pretend it is a constant.
Formula
Observation
The board moves from $\frac{\partial f}{\partial x}(1,2)$ to $\frac{\partial f}{\partial x}(x,y)$ .

Generalizes

Explanation

The general partial derivative formula extends the pointwise computation: substituting a specific point into the general formula recovers the numerical partial derivative at that point.

Holding the other variable constant → Partial derivative at a point

Clear evidence

Shown in the video

Evidence

Audio
Observation
The same “hold the other variable constant” rule is used in the pointwise $y$ -derivative and in the later general $x$ -derivative.

Application

Explanation

The method of freezing the other coordinate is the operational rule used to compute partial derivatives at a point.

Holding the other variable constant → General partial derivative as a function of $(x,y)$

Clear evidence

Shown in the video

Evidence

Audio
Observation
Speaker says that for the general formula we still pretend the other variable is a constant.

Application

Explanation

The same constant-holding principle produces the symbolic formula $\frac{\partial f}{\partial x}(x,y)=2xy+0$ .

Find an answer · 19

What is a partial derivative and why is it introduced for multivariable functions?

Clear evidence

Shown in the video

Evidence

Audio
Observation
The speaker asks how to take the derivative of a multivariable expression and names the method "partial derivative".

Knowledge points

What a partial derivative is meant to answer
Example of a scalar-valued multivariable function

What does the Leibniz notation $\frac{df}{dx}$ mean geometrically?

Clear evidence

Shown in the video

Evidence

Formula
Observation
The board shows $\frac{df}{dx}(2)$ .
Audio
Observation
The speaker explains $dx$ and $df$ and calls the ratio slope.

Knowledge points

Leibniz notation for an ordinary derivative
Graphical interpretation of $\frac{df}{dx}$ as slope

How can an ordinary derivative be understood without looking at a graph?

Clear evidence

Shown in the video

Evidence

Audio
Observation
The speaker says, "But you could also think about this without graphs if you really wanted to".
Diagram
Observation
Two number lines are drawn to represent input and output spaces.

Knowledge points

Derivative as a mapping between input and output number lines

What does $\frac{\partial f}{\partial x}$ mean for a function of two variables?

Clear evidence

Shown in the video

Evidence

Formula
Observation
The board shows $\frac{\partial f}{\partial x}$ .
Audio
Observation
The speaker interprets it as the effect of a tiny change in the input in the $x$ direction on the output.

Knowledge points

Interpretation of $\frac{\partial f}{\partial x}$ for a multivariable function

In the multivariable setup, is the drawn $xy$ -plane the graph of the function or the input space?

Clear evidence

Shown in the video

Evidence

Audio
Observation
The speaker says the plane is not graphing the function and that every point on the plane is an input.
Diagram
Observation
An $xy$ -plane is drawn as the input space.

Knowledge points

For a two-variable function, the input space is the $xy$ -plane
In the multivariable setup, the drawn plane is input space, not the graph of the function

How is the partial derivative example set up at the input point $(1,2)$ ?

Approximate timing

Shown in the video

Evidence

Formula
Observation
The board shows $(1,2)$ next to $\frac{\partial f}{\partial x}$ .
Diagram
Observation
A point is marked on the $xy$ -plane.
Audio
Observation
The speaker begins to describe a tiny change at that point but the clip ends mid-explanation.

Uncertainties

The full explanation of the tiny change at $(1,2)$ is not completed within the provided segment.

Knowledge points

Interpretation of $\frac{\partial f}{\partial x}$ for a multivariable function
For a two-variable function, the input space is the $xy$ -plane

What does a partial derivative measure for a function of several variables?

Clear evidence

Derived from the video

Evidence

Audio
Observation
The speaker explains how changing one input affects the output of a function from the plane to a number line.
Diagram
Observation
The input-plane and output-line drawing illustrates directional change.

Knowledge points

Partial derivative as directional sensitivity of a multivariable function

Why do we write $\partial f/\partial x$ instead of $df/dx$ for multivariable functions?

Clear evidence

Derived from the video

Evidence

Audio
Observation
The speaker says the curled $d$ notation emphasizes that a multivariable function is involved.
Formula
Observation
The board rewrites $df/dx$ and $df/dy$ using $\partial$ .

Knowledge points

Notation for partial derivatives

How do you evaluate $\frac{\partial f}{\partial x}(1,2)$ for $f(x,y)=x^2y+\sin(y)$ ?

Clear evidence

Derived from the video

Evidence

Formula
Observation
The worked setup substitutes $y=2$ before differentiating with respect to $x$ .
Audio
Observation
The speaker says the partial derivative treats the other variable as constant.

Knowledge points

Computing a partial derivative by holding the other variable fixed
Setting up $\frac{\partial f}{\partial x}(1,2)$ for $f(x,y)=x^2y+\sin(y)$

Does a single partial derivative describe the full change of a multivariable function?

Clear evidence

Derived from the video

Evidence

Audio
Observation
The speaker says each partial derivative is only a small part of the story of how $f$ changes.

Knowledge points

Thinking a partial derivative describes all changes of the function
Partial derivative as directional sensitivity of a multivariable function

Why can a partial derivative problem become an ordinary derivative after substitution?

Clear evidence

Derived from the video

Evidence

Audio
Observation
The speaker says that after fixing $y$ , the remaining expression is just an ordinary derivative.
Formula
Observation
The displayed expression depends only on $x$ .

Knowledge points

After freezing the other variable, the partial derivative becomes an ordinary derivative
Rewriting $\frac{\partial f}{\partial x}(1,2)$ as a one-variable derivative problem

What does it mean to compute a partial derivative at a point?

Clear evidence

Shown in the video

Evidence

Formula
Observation
Board shows both $\frac{\partial f}{\partial x}(1,2)$ and $\frac{\partial f}{\partial y}(1,2)$ .

Knowledge points

Partial derivative at a point
Worked example: partial derivatives of $f(x,y)=x^2y+\sin(y)$

Coverage and review notes

Covered · Introduction of the scalar-valued two-variable function $f(x,y)=x^2y+\sin(y)$ .

Covered · The speaker poses the differentiation question for a multivariable expression and names partial derivatives as the method.

Covered · Review of the single-variable example $f(x)=x^2$ and the notation $\frac{df}{dx}(2)$ .

Covered · Graphical explanation of $dx$ , $df$ , and slope on the parabola at $x=2$ .

Covered · Alternative non-graphical interpretation of the derivative as a map between input and output number lines.

Covered · Transfer of the derivative intuition to the multivariable case using $\frac{\partial f}{\partial x}$ .

Covered · Introduction of the $xy$ -plane as input space and marking of the point $(1,2)$ ; the final verbal elaboration is cut off but the mathematical content present in the interval is captured.

Covered · Informal explanation of how changing $x$ or $y$ separately affects the output, with matching diagrams.

Covered · Introduction of $\partial$ notation and explanation that each partial derivative captures only one direction.

Covered · The one-dimensional analogy panel is erased to make room for the worked example.

Covered · Setup of $\frac{\partial f}{\partial x}(1,2)$ by substituting $y=2$ ; the clip stops before the final derivative value is computed.

Covered · Computes $\frac{\partial f}{\partial x}(1,2)$ from the displayed function and reaches the value $4$ .

Covered · Computes $\frac{\partial f}{\partial y}(1,2)$ by holding $x=1$ fixed and obtains $1+\cos(2)$ .

Covered · Speaker transitions from point evaluations to the idea of a general formula and clears board space.

Covered · Derives the symbolic formula $\frac{\partial f}{\partial x}(x,y)=2xy+0$ and explains that it reproduces the earlier point result.

Covered · All mathematical content in the clip has been extracted and categorized.

Explore the knowledge in this video

Reviewed subject paths

Partial derivative

Partial derivatives, introduction

The explanation, unpacked.

Chapters

Learning script

Knowledge cards

Scalar-valued multivariable function example

Partial derivatives as the multivariable analogue of ordinary derivatives

Leibniz notation for an ordinary derivative

Graphical meaning of dxdxdx and dfdfdf

Derivative as amplification between input and output number lines

Meaning of ∂f∂x\frac{\partial f}{\partial x}∂x∂f​

For two-variable functions, the input space is the xyxyxy-plane

Partial derivative as change in one input direction

dxdxdx, dydydy, and dfdfdf in the visual model

Why the symbol ∂\partial∂ is used

Each partial derivative is only part of the story

Computing ∂f∂x\frac{\partial f}{\partial x}∂x∂f​ by freezing yyy

Partial derivative at a point

Worked value of ∂f∂x(1,2)\frac{\partial f}{\partial x}(1,2)∂x∂f​(1,2)

Worked value of ∂f∂y(1,2)\frac{\partial f}{\partial y}(1,2)∂y∂f​(1,2)

General partial derivative formula

Deriving ∂f∂x(x,y)\frac{\partial f}{\partial x}(x,y)∂x∂f​(x,y) for the example

Common pitfall: forgetting which variable is constant

Definition of Partial Derivative

Calculating ∂f/yf/yf/y for f(x,y)=x2y+sin⁡(y)f(x,y) = x^2y + \sin (y)f(x,y)=x2y+sin(y)

Geometric Meaning of Partial Derivatives

Limitations of Graphical Interpretation

Detailed learning notes

f(x,y)f(x,y)f(x,y)

x

y

f(x)f(x)f(x)

dfdx\frac{df}{dx}dxdf​

dx

df

x=2x=2x=2

∂f∂x\frac{\partial f}{\partial x}∂x∂f​

(1,2)

xy\text{-plane}

f(x,y)f(x,y)f(x,y)

Example of a scalar-valued multivariable function

What a partial derivative is meant to answer

Leibniz notation for an ordinary derivative

Graphical interpretation of dfdx\frac{df}{dx}dxdf​ as slope

Derivative as a mapping between input and output number lines

Interpretation of ∂f∂x\frac{\partial f}{\partial x}∂x∂f​ for a multivariable function

For a two-variable function, the input space is the xyxyxy-plane

Partial derivative as directional sensitivity of a multivariable function

Notation for partial derivatives

Computing a partial derivative by holding the other variable fixed

Partial derivative at a point

General partial derivative as a function of (x,y)(x,y)(x,y)

A partial derivative with respect to one variable ignores changes in the other variable

After freezing the other variable, the partial derivative becomes an ordinary derivative

Value of the xxx-partial at (1,2)(1,2)(1,2)

Value of the yyy-partial at (1,2)(1,2)(1,2)

General formula for the xxx-partial

Two equivalent ways to understand the ordinary derivative before introducing partial derivatives

Transferring the derivative intuition from one variable to two variables

Rewriting ∂f∂x(1,2)\frac{\partial f}{\partial x}(1,2)∂x∂f​(1,2) as a one-variable derivative problem

Derivation of ∂f∂x(1,2)\frac{\partial f}{\partial x}(1,2)∂x∂f​(1,2)

Derivation of ∂f∂y(1,2)\frac{\partial f}{\partial y}(1,2)∂y∂f​(1,2)

Derivation of the general formula for ∂f∂x(x,y)\frac{\partial f}{\partial x}(x,y)∂x∂f​(x,y)

Calculation of ∂f/yf/yf/y

Calculation of ∂f/∂x

Ordinary derivative example: f(x)=x2f(x)=x^2f(x)=x2 at x=2x=2x=2

Partial derivative setup for a two-variable function at (1,2)(1,2)(1,2)

Setting up ∂f∂x(1,2)\frac{\partial f}{\partial x}(1,2)∂x∂f​(1,2) for f(x,y)=x2y+sin⁡(y)f(x,y)=x^2y+\sin(y)f(x,y)=x2y+sin(y)

Worked example: partial derivatives of f(x,y)=x2y+sin⁡(y)f(x,y)=x^2y+\sin(y)f(x,y)=x2y+sin(y)

Evaluating Partial Derivatives at a Point

Graphical depiction of ordinary derivative as local slope

Number-line mapping interpretation of the derivative

Input-space visualization for a two-variable function

Visualizing a multivariable function as a map from the plane to a number line

Adding the yyy-direction perturbation to the same mapping picture

Replacing df/dxdf/dxdf/dx and df/dydf/dydf/dy with partial-derivative notation

Erasing the single-variable comparison panel

Writing the substitution-based setup for the partial derivative

Initial whiteboard layout

Live construction of the yyy-partial computation

Graphical meaning of $dx$ and $df$

Meaning of $\frac{\partial f}{\partial x}$

For two-variable functions, the input space is the $xy$ -plane

$dx$ , $dy$ , and $df$ in the visual model

Why the symbol $\partial$ is used

Computing $\frac{\partial f}{\partial x}$ by freezing $y$

Worked value of $\frac{\partial f}{\partial x}(1,2)$

Worked value of $\frac{\partial f}{\partial y}(1,2)$

Deriving $\frac{\partial f}{\partial x}(x,y)$ for the example

Calculating ∂ $f/y$ for $f(x,y) = x^2y + \sin (y)$

$f(x,y)$

$f(x)$

$\frac{df}{dx}$

$x=2$

$\frac{\partial f}{\partial x}$

$f(x,y)$

Graphical interpretation of $\frac{df}{dx}$ as slope

Interpretation of $\frac{\partial f}{\partial x}$ for a multivariable function

For a two-variable function, the input space is the $xy$ -plane

General partial derivative as a function of $(x,y)$

Value of the $x$ -partial at $(1,2)$

Value of the $y$ -partial at $(1,2)$

General formula for the $x$ -partial

Rewriting $\frac{\partial f}{\partial x}(1,2)$ as a one-variable derivative problem

Derivation of $\frac{\partial f}{\partial x}(1,2)$

Derivation of $\frac{\partial f}{\partial y}(1,2)$

Derivation of the general formula for $\frac{\partial f}{\partial x}(x,y)$

Calculation of ∂ $f/y$

Ordinary derivative example: $f(x)=x^2$ at $x=2$

Partial derivative setup for a two-variable function at $(1,2)$

Setting up $\frac{\partial f}{\partial x}(1,2)$ for $f(x,y)=x^2y+\sin(y)$

Worked example: partial derivatives of $f(x,y)=x^2y+\sin(y)$

Adding the $y$ -direction perturbation to the same mapping picture

Replacing $df/dx$ and $df/dy$ with partial-derivative notation

Live construction of the $y$ -partial computation

Writing the general $x$ -partial formula

Leibniz notation for an ordinary derivative → Interpretation of $\frac{\partial f}{\partial x}$ for a multivariable function

Graphical interpretation of $\frac{df}{dx}$ as slope → Derivative as a mapping between input and output number lines

Derivative as a mapping between input and output number lines → For a two-variable function, the input space is the $xy$ -plane

Interpretation of $\frac{\partial f}{\partial x}$ for a multivariable function → For a two-variable function, the input space is the $xy$ -plane

Computing a partial derivative by holding the other variable fixed → Setting up $\frac{\partial f}{\partial x}(1,2)$ for $f(x,y)=x^2y+\sin(y)$

Partial derivative as directional sensitivity of a multivariable function → $f(x)=x^2$

General partial derivative as a function of $(x,y)$ → Partial derivative at a point

Holding the other variable constant → General partial derivative as a function of $(x,y)$

What does the Leibniz notation $\frac{df}{dx}$ mean geometrically?

What does $\frac{\partial f}{\partial x}$ mean for a function of two variables?

In the multivariable setup, is the drawn $xy$ -plane the graph of the function or the input space?

How is the partial derivative example set up at the input point $(1,2)$ ?

Why do we write $\partial f/\partial x$ instead of $df/dx$ for multivariable functions?

How do you evaluate $\frac{\partial f}{\partial x}(1,2)$ for $f(x,y)=x^2y+\sin(y)$ ?