Directional derivative

1. Formula
   Observation

   At 00:05.000 the board shows $f(x,y)$.

2. Audio
   Observation

   The speaker refers to a function with two inputs and a single-variable real-number output.

f

A scalar-valued multivariable function used as the example object for discussing derivatives.

Function of two variables; in this clip the speaker restricts attention to a single real-number output.

1. Formula
   Observation

   At 00:05.000 x appears inside $f(x,y)$.

2. Diagram
   Observation

   At 00:32.000 the horizontal axis is labeled x.

x

First input variable of f and coordinate along the horizontal axis of the input plane.

Real coordinate in the input space.

1. Formula
   Observation

   At 00:05.000 y appears inside $f(x,y)$.

2. Diagram
   Observation

   At 00:32.000 the vertical axis is labeled y.

y

Second input variable of f and coordinate along the vertical axis of the input plane.

Real coordinate in the input space.

1. Formula
   Observation

   At 00:53.000 the board shows ∂f/∂$x(1,2)$.

2. Audio
   Observation

   The speaker says partial derivative of f with respect to x at a point like one two.

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

Partial derivative of f with respect to x evaluated at the input point (1,2).

Scalar value associated with nudging the input in the x-direction at (1,2).

1. Formula
   Observation

   At 01:33.000 the board shows $\vec{v} = [-1, 2]$.

2. Audio
   Observation

   The speaker introduces some vector v and says let's say it's negative one two.

$\vec{v}$

A chosen direction vector in the input plane.

Two-dimensional vector; here explicitly given as [-1, 2].

1. Formula
   Observation

   At 02:35.000 the board shows $h\vec{v}$.

2. Audio
   Observation

   The speaker says you'd be thinking of taking a step along say h multiplied by that vector.

h

A very small scalar multiplier used to turn the direction vector into an infinitesimal step.

Scalar approaching 0 in the limiting discussion.

1. Formula
   Observation

   $f(x,y) = x^2y$

$f(x,y)$

Scalar function of two variables

$R^2$

1. Formula
   Observation

   $v = [-1, 2]$

v

Specific direction vector with components -1 and 2

$R^2$

1. Formula
   Observation

   $h v = [-h, 2h]$

h

Small scalar step size

R

1. Formula
   Observation

   $w = [a, b]$

w

General direction vector with abstract components a and b

$R^2$

1. Formula
   Observation

   $w = [a, b]$

a, b

Components of the general direction vector w

R

1. Formula
   Observation

   ∇_v $f(x,y)$

∇_v f

Directional derivative of f in the direction of v

$R^2$ -> R

1. Formula
   Observation

   At 00:05.000 the board writes $f(x,y)$.

2. Audio
   Observation

   From 00:05.000 to 00:32.000 the speaker says partial derivatives have to do with functions with some kind of multivariable input, uses two inputs because that's easiest, and focuses on a single-variable ordinary real number output.

The clip begins by fixing the setting for directional derivatives: a function with two input variables and one real-number output. The speaker explicitly narrows the discussion from possible vector-valued outputs to a scalar-valued function so the geometric picture stays simple.

1. Two input variables are used for simplicity.

2. The output is treated as a single real number in this clip.

3. Vector-valued outputs are mentioned but not developed here.

1. Diagram
   Observation

   At 00:32.000 a coordinate plane labeled x and y is drawn on the left.

2. Diagram
   Observation

   At 00:39.000 a separate horizontal line labeled f is drawn on the right.

3. Audio
   Observation

   From 00:32.000 to 00:53.000 the speaker describes the input space as the x and y plane and says this outputs to just the real numbers.

The function is visualized as a map from an input plane to an output line. The left side represents the domain coordinates (x,y), while the right side represents the scalar output space labeled f.

1. Used as a geometric interpretation of a scalar-valued function of two variables.

1. Multivariable scalar function setup

1. Formula
   Observation

   At 00:53.000 the board shows ∂$f/x(1,2)$.

2. Diagram
   Observation

   At 01:07.000 a yellow arrow points horizontally right from the point (1,2) in the input plane.

3. Diagram
   Observation

   At 01:10.000 a yellow arrow appears on the output line.

4. Audio
   Observation

   From 00:53.000 to 01:20.000 the speaker explains nudging in the x direction and taking the ratio between the resulting output nudge and the original one.

5. Audio
   Observation

   From 01:20.000 to 01:33.000 the speaker contrasts this with moving straight up in the y direction.

The partial derivative is presented operationally: choose a point in the input plane, make a tiny displacement along one coordinate direction, and compare the induced change in the output. For ∂f/∂x the displacement is horizontal; for the y-partial it is vertical.

1. Evaluated at a specific input point, here (1,2).

2. The displacement is along a coordinate axis direction.

3. The idea is based on an infinitesimal nudge rather than a finite step.

1. Multivariable scalar function setup
2. Input plane and output line representation

1. Formula
   Observation

   At 01:33.000 the board writes $\vec{v} = [-1, 2]$.

2. Diagram
   Observation

   At 01:51.000 a purple arrow is drawn from the point (1,2) toward the upper-left.

3. Audio
   Observation

   From 01:33.000 to 01:58.000 the speaker says what if you have some vector v ... negative one two ... a step of negative one in the x direction and then two more in the y direction.

The lecture generalizes from axis-aligned nudges to a chosen direction in the input plane. The vector $\vec{v}=[-1,2]$ is used as a concrete example of a non-axis direction, interpreted as moving left by 1 and up by 2 from the base point.

1. The vector is attached to the point (1,2) in the visual explanation.

2. It represents a direction in the input space, not yet the infinitesimal step itself.

1. Partial derivative as a directional nudge along coordinate axes

1. Audio
   Observation

   From 01:58.000 to 02:02.000 the speaker asks what does a nudge in that direction do to the function itself.

2. Audio
   Observation

   From 02:02.000 to 02:28.000 the speaker stresses that the relevant step is really something very very small and formally involves a limit approaching zero.

3. Formula
   Observation

   At 02:35.000 the board writes $h\vec{v}$.

4. Audio
   Observation

   From 02:28.000 to 02:51.000 the speaker says you'd be thinking of taking a step along say h multiplied by that vector, where h might represent some really really small number, and formally the limit as h goes to zero.

5. Audio
   Observation

   From 02:51.000 to 02:58.000 the speaker summarizes: when you take a slight nudge in the direction of that vector, what is the resulting change to the output?

The directional derivative is introduced as the analogue of a partial derivative but for an arbitrary direction vector. Instead of moving exactly by $\vec{v}$, one considers an infinitesimal step $h\vec{v}$ with $h\to 0$ and asks how the output changes relative to that tiny input displacement.

1. The direction is specified by a vector $\vec{v}$.

2. The actual displacement considered is scaled by a small scalar h.

3. The formal idea uses the limit as h approaches 0.

4. This clip gives the concept and setup, not a completed formula or worked computation.

1. Partial derivative as a directional nudge along coordinate axes
2. Introducing an arbitrary direction vector

1. Audio
   Observation

   slight nudge of the vector if we actually expand things out... negative h, negative one times that component, and then 2h here.

2. Formula
   Observation

   $h v = [-h, 2h]$

The directional derivative can be understood by taking a small step (nudge) in the direction of a vector. For a vector $v = [-1, 2]$, a step of size h results in a displacement of [-h, 2h], meaning a change of -h in the x-direction and +2h in the y-direction.

1. v is a direction vector

2. h is a small scalar step

1. Audio
   Observation

   notation by the way is um you take that same nabla from the gradient but then you put the vector down here... whole bunch of other notations too... partial with a little subscript vector

2. Formula
   Observation

   ∇_v f, ∂f/∂v, ∂_v f

There are multiple notations for the directional derivative. The speaker prefers ∇_v $f(x,y)$, using the nabla symbol with the direction vector as a subscript. Other common notations include ∂$f/v$ and ∂_v f.

1. Definition of Directional Derivative via Nudge

1. Audio
   Observation

   good guess that you might have is to say well we take a negative step in the x direction... do the negative of that... two steps in the y direction... two times partial f partial y

2. Formula
   Observation

   ∇_v $f(x,y)$ = -∂$f/x + 2$∂f/∂y

To calculate the directional derivative for a specific vector, multiply each partial derivative by the corresponding component of the direction vector. For $v = [-1, 2]$, this means taking -1 times the partial derivative with respect to x, plus 2 times the partial derivative with respect to y.

1. $v = [-1, 2]$

1. Definition of Directional Derivative via Nudge
2. ∂$f/x$
3. ∂$f/y$

1. Audio
   Observation

   let's say we've got a vector w... call it a and b... directional derivative in the direction of w... is equal to a times the partial derivative of f with respect to x plus b times the partial derivative of f with respect to y

2. Formula
   Observation

   ∇_w $f = a$ ∂$f/x + b$ ∂f/∂y

For a general direction vector $w = [a, b]$, the directional derivative is the dot product of the vector components and the partial derivatives: a * (∂f/∂x) + b * (∂f/∂y).

1. $w = [a, b]$

1. Calculating Directional Derivative from Components
2. w

1. Audio
   Observation

   sometimes you see this written... with respect to the gradient... makes it much more compact... looks like a dot product... dot product of the vectors a b and the one that has the partial derivatives in it

2. Formula
   Observation

   ∇_w $f = [a, b]$ · [∂$f/x$, ∂f/∂y]

The general formula for the directional derivative can be expressed compactly as the dot product of the direction vector $w = [a, b]$ and the gradient vector ∇f = [∂f/∂x, ∂f/∂y]. This form is more general and works for higher dimensions.

1. $w = [a, b]$

2. ∇f is the gradient vector

1. General Formula for Directional Derivative
2. ∇f

1. Formula
   Observation

   Red box: $\nabla_{\vec w} f = a\frac{\partial f}{\partial x} + b\frac{\partial f}{\partial y}$.

2. Audio
   Observation

   Speaker explains that the coefficients come from the vector components and the partials from the gradient.

For a scalar field $f(x,y)$ and a direction vector $\vec w=\begin{bmatrix}a\\b\end{bmatrix}$, the directional derivative shown on the board is the weighted sum of the partial derivatives, with weights equal to the components of $\vec w$.

1. $f$ is a differentiable scalar field of two variables in the context shown.

2. $\vec w$ is treated as the direction vector with components $a,b$.

3. The video does not explicitly state whether $\vec w$ must be normalized.

1. Gradient vector of a two-variable function
2. Partial derivatives as components of the gradient

1. Formula
   Observation

   Column vector $\begin{bmatrix}\frac{\partial f}{\partial x}\\\frac{\partial f}{\partial y}\end{bmatrix}$ is written next to $\vec w$.

2. Audio
   Observation

   Speaker says, "That's just the gradient. That is the gradient of f."

The gradient of $f$ is the vector whose components are the partial derivatives of $f$ with respect to each input variable. In two dimensions it is written as a column vector of $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$.

1. $f$ has partial derivatives with respect to its input variables.

2. The displayed form is for a function of two variables.

1. Partial derivatives as components of the gradient

1. Formula
   Observation

   Board shows both $\nabla_{\vec w} f = a\frac{\partial f}{\partial x} + b\frac{\partial f}{\partial y}$ and $\vec w\cdot\nabla f$.

2. Audio
   Observation

   Speaker identifies the column of partials as the gradient and says this is the compact textbook notation.

In the two-variable setting shown, $\nabla_{\vec w} f = a\frac{\partial f}{\partial x} + b\frac{\partial f}{\partial y}$ is equivalent to $\vec w\cdot\nabla f$ when $\vec w=\begin{bmatrix}a\\b\end{bmatrix}$ and $\nabla f=\begin{bmatrix}\frac{\partial f}{\partial x}\\\frac{\partial f}{\partial y}\end{bmatrix}$.

1. $f$ is a scalar field of two variables.

2. $\vec w$ has components $a,b$.

3. The gradient is formed from the corresponding partial derivatives.

For the displayed two-variable setup.

1. Audio
   Observation

   Speaker says, "if we were talking about something that has like a five dimensional input ... the gradient would have five components and the vector itself would have five components."

1. The video states this verbally but does not write the five-dimensional formula on screen.

The compact notation $\vec w\cdot\nabla f$ remains flexible in higher dimensions; for a five-dimensional input, both the direction vector and the gradient have five components.

1. The function has a higher-dimensional input.

2. The directional derivative is expressed using the same dot-product structure.

Illustrated by a five-dimensional example.

1. Audio
   Observation

   From 00:53.000 to 01:33.000 the speaker explains partial derivatives as nudges in x and y directions.

2. Audio
   Observation

   From 01:33.000 to 02:58.000 the speaker generalizes to an arbitrary vector direction and introduces $h\vec{v}$ with $h\to 0$.

3. Diagram
   Observation

   At 01:07.000 and 01:25.000 axis-aligned arrows are shown; at 01:51.000 a slanted vector is shown from the same base point.

1. Expression

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

   Start with the partial derivative at the point (1,2) in the x-direction.

   Justification

   Directly stated in the audio and written on the board at 00:53.000.

   Shown in the video
2. Expression

   $$\text{nudge in }x\text{ direction} \Rightarrow \text{output change}$$
   Explanation

   Interpret the partial derivative as comparing a tiny input displacement along x with the resulting displacement in the output line.

   Justification

   Spoken explanation from 00:53.000 to 01:20.000, supported by the yellow arrows at 01:07.000 and 01:10.000.

   Shown in the video
3. Expression

   $$\text{nudge in }y\text{ direction} \Rightarrow \text{different output change}$$
   Explanation

   Repeat the same idea for the y-direction to show that the direction of the input nudge matters.

   Justification

   Spoken contrast from 01:20.000 to 01:33.000, supported by the red upward arrow at 01:25.000.

   Shown in the video
4. Expression

   $$\vec{v} = \begin{bmatrix}-1\\2\end{bmatrix}$$
   Explanation

   Replace the coordinate directions with an arbitrary direction vector in the input plane.

   Justification

   Introduced verbally and visually at 01:33.000 and 01:51.000.

   Shown in the video
5. Expression

   $$h\vec{v},\quad h\to 0$$
   Explanation

   Do not use the full vector as the step; instead scale it by a very small scalar h and take the limit as h goes to zero.

   Justification

   Explicitly stated from 02:28.000 to 02:51.000 and written on the board at 02:35.000.

   Shown in the video
6. Expression

   $$\text{directional derivative} = \text{resulting output change for a slight nudge in the direction of }\vec{v}$$
   Explanation

   Conclude that the directional derivative measures the infinitesimal output change caused by moving in the chosen direction.

   Justification

   Summary sentence from 02:51.000 to 02:58.000.

   Shown in the video

The clip motivates the directional derivative by generalizing the partial-derivative nudge picture from the coordinate axes to an arbitrary vector direction, with the infinitesimal step represented as $h\vec{v}$ and h tending to 0.

1. Audio
   Observation

   And this is actually how you calculate it. And if I was going to be more general...

2. Formula
   Observation

   Transition from ∇_v f = -∂f/∂$x + 2$∂f/∂y to _w $f = a f$/∂$x + b$ ∂f/∂y

1. Expression

   $$\nabla_{\vec{v}} f(x,y) = -\frac{\partial f}{\partial x} + 2\frac{\partial f}{\partial y}$$
   Explanation

   Start with the calculation for the specific vector $v = [-1, 2]$.

   Justification

   Based on the components of v acting as multipliers for the partial derivatives.

   Shown in the video
2. Expression

   $$\vec{w} = \begin{bmatrix} a \\ b \end{bmatrix}$$
   Explanation

   Introduce a general vector w with abstract components a and b.

   Justification

   To generalize the result beyond the specific numbers -1 and 2.

   Shown in the video
3. Expression

   $$\nabla_{\vec{w}} f = a \frac{\partial f}{\partial x} + b \frac{\partial f}{\partial y}$$
   Explanation

   Replace the specific components -1 and 2 with general components a and b.

   Justification

   The structure of the calculation remains the same: component times corresponding partial derivative.

   Shown in the video

The directional derivative in the direction of any vector $w=[a,b]$ is given by the linear combination of partial derivatives weighted by the vector's components.

1. Audio
   Observation

   If you look at this expression, it looks like a dot product.

2. Formula
   Observation

   ∇_w $f = [a, b]$ · [∂$f/x$, ∂$f/y$]

1. Expression

   $$\nabla_{\vec{w}} f = a \frac{\partial f}{\partial x} + b \frac{\partial f}{\partial y}$$
   Explanation

   Begin with the general algebraic formula for the directional derivative.

   Justification

   Established in the previous derivation.

   Shown in the video
2. Expression

   $$\begin{bmatrix} a \\ b \end{bmatrix} \cdot \begin{bmatrix} \frac{\partial f}{\partial x} \\ \frac{\partial f}{\partial y} \end{bmatrix}$$
   Explanation

   Recognize the sum of products as the definition of a dot product between two vectors.

   Justification

   The first vector contains the components of the direction, and the second contains the partial derivatives (the gradient).

   Shown in the video

The directional derivative can be computed as the dot product of the direction vector and the gradient vector.

1. Formula
   Observation

   The board shows $\begin{bmatrix}a\\b\end{bmatrix}\cdot\begin{bmatrix}\frac{\partial f}{\partial x}\\\frac{\partial f}{\partial y}\end{bmatrix}$ and then the compact form $\vec w\cdot\nabla f$.

2. Audio
   Observation

   Speaker walks through matching $a,b$ to the vector and the partials to the gradient.

1. Expression

   $$\nabla_{\vec w} f = a\frac{\partial f}{\partial x} + b\frac{\partial f}{\partial y}$$
   Explanation

   Start from the directional derivative formula already written in the red box.

   Justification

   Observed directly on the board.

   Shown in the video
2. Expression

   $$\begin{bmatrix}a\\b\end{bmatrix}\cdot\begin{bmatrix}\frac{\partial f}{\partial x}\\\frac{\partial f}{\partial y}\end{bmatrix}$$
   Explanation

   Recognize the coefficients $a,b$ as the components of $\vec w$ and the two partial derivatives as the components of a vector.

   Justification

   Audio identifies $a,b$ as the original vector and the partials as the gradient.

   Shown in the video
3. Expression

   $$\vec w\cdot\nabla f$$
   Explanation

   Replace the component vector by $\vec w$ and the partial-derivative vector by $\nabla f$.

   Justification

   Definition of $\vec w$ and gradient notation.

   Shown in the video

The expanded directional derivative is equivalently written as the dot product of the direction vector with the gradient.

1. Formula
   Observation

   At 00:53.000 the board shows ∂f/∂$x(1,2)$.

2. Diagram
   Observation

   At 01:01.000 the point (1,2) is marked in the input plane.

3. Formula
   Observation

   At 01:33.000 the board shows $\vec{v} = [-1, 2]$.

4. Diagram
   Observation

   At 01:51.000 a purple vector is drawn from (1,2).

5. Formula
   Observation

   At 02:35.000 the board shows $h\vec{v}$.

1. No numerical value for the directional derivative is computed in this clip.

2. The example remains conceptual; the function rule for f is not specified beyond the symbolic form $f(x,y)$.

Explain how a partial derivative at (1,2) is visualized and how the same idea extends to a directional derivative using the vector $\vec{v}=[-1,2]$.

1. A scalar function $f(x,y)$.

2. Evaluation point (1,2) in the input plane.

3. Direction vector $\vec{v}=[-1,2]$.

4. Small scalar multiplier h with $h\to 0$.

Describe the geometric meaning of the partial derivative and the directional derivative at the chosen point and direction.

1. Expression

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

   Mark the point (1,2) and consider a tiny horizontal displacement in the x-direction.

   Justification

   Shown on the board at 00:53.000 and illustrated by the yellow rightward arrow at 01:07.000.

   Shown in the video
2. Expression

   $$\text{output-line arrow}$$
   Explanation

   Show the corresponding tiny change on the output line labeled f.

   Justification

   Drawn at 01:10.000 and described in the audio as the resulting nudge in the output space.

   Shown in the video
3. Expression

   $$\vec{v} = \begin{bmatrix}-1\\2\end{bmatrix}$$
   Explanation

   Introduce a non-axis direction by drawing a vector from (1,2) that moves left 1 and up 2.

   Justification

   Written at 01:33.000 and drawn at 01:51.000.

   Shown in the video
4. Expression

   $$h\vec{v}$$
   Explanation

   Replace the full vector step with an infinitesimal scaled step in the same direction.

   Justification

   Written at 02:35.000 and explained verbally from 02:28.000 to 02:51.000.

   Shown in the video
5. Expression

   $$h\to 0$$
   Explanation

   Take the limit as the scalar step size shrinks to zero to define the directional rate of change.

   Justification

   Stated verbally from 02:14.000 to 02:51.000.

   Shown in the video

The example shows that ∂f/∂$x(1,2)$ corresponds to an infinitesimal horizontal input nudge and its induced output change, while the directional derivative generalizes this to an infinitesimal nudge in the direction of $\vec{v}$ via $h\vec{v}$ with $h\to 0$.

Verification is geometric and verbal only: the board consistently marks the base point (1,2), draws the axis-aligned and slanted directions from that point, and the narration matches the drawn input-output picture. No algebraic check is performed in the clip.

1. Formula
   Observation

   $f(x,y) = x^2y$, $v = [-1, 2]$

2. Audio
   Observation

   negative h, negative one times that component, and then 2h here... good guess... negative step in the x direction... two steps in the y direction

Find the expression for the directional derivative of $f(x,y)=x^2y$ in the direction of $v=[-1, 2]$.

1. $f(x,y) = x^2y$

2. $v = [-1, 2]$

Express ∇_v f in terms of partial derivatives.

1. Expression

   $$h\vec{v} = \begin{bmatrix} -h \\ 2h \end{bmatrix}$$
   Explanation

   Calculate the small displacement vector by scaling v by h.

   Justification

   Definition of scalar multiplication of a vector.

   Shown in the video
2. Expression

   $$\nabla_{\vec{v}} f(x,y) = -\frac{\partial f}{\partial x} + 2\frac{\partial f}{\partial y}$$
   Explanation

   Multiply the partial derivative with respect to x by the x-component (-1) and the partial derivative with respect to y by the y-component (2), then sum them.

   Justification

   Method for calculating directional derivative from vector components.

   Shown in the video

-\frac{$\partial f$}{$\partial x$} + $2\frac{\partial f}{\partial y}$

Matches the general formula when $a=-1$ and $b=2$.

1. Formula
   Observation

   Top-center vector is $\vec v=\begin{bmatrix}-1\\2\end{bmatrix}$.

2. Formula
   Observation

   Yellow formula reads $\nabla_{\vec v} f(x,y) = -\frac{\partial f}{\partial x} + 2\frac{\partial f}{\partial y}$.

1. The clip does not evaluate this expression numerically at a point.

Given $f(x,y)=x^2y$ and $\vec v=\begin{bmatrix}-1\\2\end{bmatrix}$, express the directional derivative in the direction of $\vec v$.

1. $f(x,y)=x^2y$

2. $\vec v=\begin{bmatrix}-1\\2\end{bmatrix}$

Write $\nabla_{\vec v} f(x,y)$ in terms of the partial derivatives.

1. Expression

   $$\vec v = \begin{bmatrix}-1\\2\end{bmatrix}$$
   Explanation

   Identify the components of the direction vector.

   Justification

   Shown on the board.

   Shown in the video
2. Expression

   $$\nabla_{\vec v} f(x,y) = (-1)\frac{\partial f}{\partial x} + (2)\frac{\partial f}{\partial y}$$
   Explanation

   Substitute the components of $\vec v$ into the general directional-derivative pattern.

   Justification

   Uses the same coefficient-by-component rule shown for $\vec w$.

   Derived from the video
3. Expression

   $$\nabla_{\vec v} f(x,y) = -\frac{\partial f}{\partial x} + 2\frac{\partial f}{\partial y}$$
   Explanation

   Simplify the coefficients.

   Justification

   Algebraic simplification.

   Shown in the video

$\nabla_{\vec v} f(x,y) = -\frac{\partial f}{\partial x} + 2\frac{\partial f}{\partial y}$

Matches the yellow formula written on the board.

1. Formula
   Observation

   Red annotation shows $h\vec v = \begin{bmatrix}-h\\2h\end{bmatrix}$ with $h=0.001$ nearby.

2. Audio
   Observation

   Speaker describes moving along the vector by a tiny nudge and asking how the output changes.

1. The clip does not compute the resulting change in $f$ or the ratio explicitly.

Interpret the meaning of multiplying a direction vector by a small scalar step.

1. $\vec v=\begin{bmatrix}-1\\2\end{bmatrix}$

2. $h=0.001$

Express the small displacement vector and connect it to the idea of a directional derivative.

1. Expression

   $$h\vec v = 0.001\begin{bmatrix}-1\\2\end{bmatrix}$$
   Explanation

   Scale the direction vector by a small positive number.

   Justification

   Shown on the board.

   Shown in the video
2. Expression

   $$h\vec v = \begin{bmatrix}-h\\2h\end{bmatrix}$$
   Explanation

   Distribute the scalar into each component.

   Justification

   Componentwise scalar multiplication.

   Shown in the video

The small displacement is $\begin{bmatrix}-h\\2h\end{bmatrix}$, representing a tiny move in the direction of $\vec v$.

Consistent with the speaker's verbal interpretation of moving along the vector by a tiny nudge.

1. Diagram
   Observation

   At 00:32.000 a blue coordinate plane with axes labeled x and y is drawn on the left.

2. Diagram
   Observation

   At 00:39.000 a separate horizontal blue line labeled f is drawn on the right.

3. Diagram
   Observation

   At 00:44.000 a curved blue arrow connects the input plane to the output line.

1. x-axis

2. y-axis

3. input plane

4. output line labeled f

5. curved mapping arrow

1. The input space is drawn first.

2. The output space is added second.

3. A connecting arrow is drawn last to indicate mapping from inputs to outputs.

1. The left object represents the two-variable input space.

2. The right object represents the one-dimensional output space.

The picture encodes f as a transformation from a point (x,y) in the plane to a real number on the output line.

1. Diagram
   Observation

   At 01:01.000 the point (1,2) is marked in the input plane.

2. Diagram
   Observation

   At 01:07.000 a yellow arrow points horizontally right from that point.

3. Diagram
   Observation

   At 01:10.000 a yellow arrow appears on the output line.

1. point (1,2)

2. yellow horizontal input arrow

3. yellow output arrow

1. A base point is selected in the input plane.

2. A rightward input displacement is shown.

3. A corresponding displacement on the output line is shown.

1. The input displacement is aligned with the x-axis.

2. The output displacement remains on the one-dimensional output line.

The paired arrows illustrate that changing only x produces a corresponding change in f, which is the geometric meaning of ∂f/∂x at that point.

1. Diagram
   Observation

   At 01:25.000 a red arrow points vertically upward from the point (1,2).

2. Diagram
   Observation

   At 01:28.000 a red arrow appears on the output line.

1. point (1,2)

2. red vertical input arrow

3. red output arrow

1. The direction of the input nudge changes from horizontal to vertical.

2. A new output displacement is indicated in red.

1. The base point remains (1,2).

2. The output remains represented on the same line.

This visual contrast shows that a different input direction yields a different induced output change, distinguishing the y-partial from the x-partial.

1. Formula
   Observation

   At 01:33.000 the board writes $\vec{v} = [-1, 2]$.

2. Diagram
   Observation

   At 01:51.000 a purple arrow is drawn from (1,2) toward the upper-left.

1. formula $\vec{v} = [-1,2]$

2. purple vector arrow

3. base point (1,2)

1. A new symbolic vector is introduced.

2. A slanted arrow is drawn from the same base point used for the partial derivatives.

1. The vector is anchored at (1,2).

2. The direction is no longer aligned with either coordinate axis.

The purple arrow concretizes the idea of moving in an arbitrary direction in the input plane, setting up the directional derivative.

1. Formula
   Observation

   At 02:35.000 the board writes $h\vec{v}$.

2. Audio
   Observation

   From 02:28.000 to 02:51.000 the speaker says h might represent some really really small number and mentions the limit as h goes to zero.

1. No explicit limit expression is fully written on the board within this clip.

2. The numeric example 0.001 is spoken but not clearly established as a written board item in the available visual evidence.

1. symbol h

2. vector $\vec{v}$

3. product $h\vec{v}$

1. The discussion shifts from the full direction vector to a scaled version.

2. The scalar h is identified as very small and tending to zero.

1. The direction remains that of $\vec{v}$.

2. The construction is still an input-space displacement used to infer output change.

Writing $h\vec{v}$ signals that the directional derivative concerns an infinitesimal step in the direction of $\vec{v}$, not the finite vector itself.

1. Diagram
   Observation

   Black background with handwritten equations and diagrams in various colors (green, purple, red, yellow, blue).

1. Coordinate system with vectors

2. Function definition $f(x,y)=x^2y$

3. Vector v definition

4. Scaled vector hv calculation

5. Directional derivative notation

6. General vector w definition

7. General formula

8. Dot product representation

1. Writing appears sequentially as the speaker explains concepts.

2. Arrows connect related concepts (e.g., from input space to output space).

3. Colors distinguish different mathematical objects (vectors, scalars, operators).

1. The core function $f(x,y)=x^2y$ remains visible throughout.

2. The initial setup of the coordinate system and vector v stays on screen.

The visual progression mirrors the logical development from a specific numerical example to a general algebraic formula, and finally to a geometric interpretation using the gradient.

1. Diagram
   Observation

   Static blackboard-style layout with function at top left, vectors at top center/right, formulas across middle, and compact notation added at bottom right.

1. Coordinate axes

2. Function $f(x,y)=x^2y$

3. Vectors $\vec v$ and $\vec w$

4. Directional derivative formulas

5. Gradient column vector

6. Compact dot-product notation

1. New writing appears at the bottom right during the clip: first $\vec w$, then $\vec w\cdot\nabla f$.

1. The main formulas and diagrams already on the board remain visible throughout the clip.

The visual progression emphasizes rewriting the expanded directional derivative in compact gradient-dot-product form.

1. Diagram
   Observation

   A curved blue arrow points from the upper-left vector diagram toward a horizontal line labeled $f$ with red and yellow arrows on it.

1. The exact semantic mapping of the red/yellow arrows on the horizontal line is not fully explained in this clip.

1. Curved blue arrow

2. Horizontal line labeled $f$

3. Red arrow

4. Yellow arrow

1. The arrow visually connects movement in the input plane to change along the output line.

1. The line and arrows remain static once shown.

This supports the speaker's later verbal idea of moving along a direction and observing how the output changes.

1. Animation
   Observation

   At the bottom right, new purple writing appears progressively: first $\vec w$, then the dot, then $\nabla f$.

1. $\vec w$

2. $\cdot$

3. $\nabla f$

1. The compact expression is built piece by piece rather than appearing all at once.

1. The surrounding formulas stay unchanged while the new notation is added.

The animation highlights the transition from componentwise expression to the concise dot-product form.

1. Audio
   Observation

   From 02:02.000 to 02:28.000 the speaker says you're not really thinking of it as a large step; you're really thinking of it as something very very small.

2. Audio
   Observation

   From 02:28.000 to 02:35.000 the speaker says you're not thinking of the actual vector actually taking a step along that.

One might think the directional derivative means literally moving by the full vector $\vec{v}$.

The clip stresses that $\vec{v}$ gives the direction, while the relevant displacement is a very small scaled step $h\vec{v}$ with $h\to 0$.

1. Diagram
   Observation

   At 01:07.000 and 01:25.000 arrows show input nudges along x and y.

2. Audio
   Observation

   From 00:53.000 to 01:33.000 the speaker repeatedly describes partial derivatives in terms of nudging the input and observing the output change.

One might regard ∂f/∂x and the y-partial only as formulas, missing that they correspond to specific coordinate directions in the input space.

The video presents partial derivatives as direction-specific infinitesimal changes: horizontal for x and vertical for y.

1. Audio
   Observation

   There's a whole bunch of other notations too... some people will just write like partial with a little subscript vector... but this is the one I like.

Students might be confused seeing different symbols (∇_v, ∂/∂v, ∂_v) for the same concept in different textbooks or lectures.

The speaker acknowledges multiple valid notations exist. The key is understanding they all represent the rate of change in a specific direction, regardless of the symbol used.

1. Audio
   Observation

   Speaker says, "here ... it's nabla without that little w at the bottom."

One might think the plain gradient symbol and the directional-derivative symbol are the same object.

The video distinguishes $\nabla_{\vec w} f$ (directional derivative in direction $\vec w$) from $\nabla f$ (the gradient vector itself); the compact formula uses the dot product of the two.

1. Audio
   Observation

   Speaker explicitly discusses a five-dimensional input as an example of flexibility.

Because the board shows only $x$ and $y$, one may infer the method is limited to two variables.

The speaker states that the dot-product formulation generalizes; in five dimensions, both the direction vector and gradient simply have five components.

1. Audio
   Observation

   From 00:01.000 to 00:05.000 the speaker says the directional derivative extends the idea of a partial derivative.

2. Audio
   Observation

   From 01:33.000 onward the speaker replaces coordinate-axis nudges with an arbitrary vector direction.

The directional derivative is presented as a generalization of the partial derivative from coordinate-axis directions to an arbitrary vector direction.

1. Formula
   Observation

   At 00:05.000 $f(x,y)$ is written before any derivative notation appears.

2. Audio
   Observation

   From 00:05.000 to 00:53.000 the speaker first defines the function setting and only then discusses partial derivatives.

Understanding the partial derivative in this clip depends on first viewing f as a scalar function of two input variables.

1. Diagram
   Observation

   At 00:32.000 and 00:39.000 the input plane and output line are drawn before the derivative arrows.

2. Diagram
   Observation

   At 01:07.000 and 01:10.000 the partial-derivative explanation uses those same two spaces.

The input-plane/output-line representation is the visual framework used to explain what a partial derivative does.

1. Formula
   Observation

   At 01:33.000 $\vec{v} = [-1,2]$ is introduced.

2. Audio
   Observation

   From 01:58.000 to 02:58.000 the speaker asks what a nudge in that direction does and then defines the infinitesimal step $h\vec{v}$.

The directional derivative concept in this clip is built directly on the prior introduction of an arbitrary direction vector in the input plane.

1. Formula
   Observation

   ∇_w $f = a$ ∂f/∂$x + b$ ∂f/∂y

The general formula for the directional derivative is constructed by applying the partial derivatives of the function.

1. Formula
   Observation

   ∇_w $f = w$ · ∇f

The directional derivative is equivalent to the dot product of the direction vector and the gradient vector.

1. Audio
   Observation

   if I was going to be more general...

The formula using abstract components a and b generalizes the calculation method shown for the specific vector [-1, 2].

1. Formula
   Observation

   $\nabla f$ is written as the column vector of $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$.

The gradient is assembled from the partial derivatives.

1. Formula
   Observation

   The compact notation $\vec w\cdot\nabla f$ is introduced after identifying the partial-derivative vector as the gradient.

Recognizing the partial-derivative vector as $\nabla f$ is what allows the directional derivative to be rewritten as a dot product.

1. Formula
   Observation

   The board shows the general formula with $\vec w=(a,b)$ and the specific formula with $\vec v=(-1,2)$.

The example with $\vec v=(-1,2)$ is a direct substitution instance of the general directional-derivative formula.

1. Audio
   Observation

   Speaker says the notation is more flexible for dimensions and gives a five-dimensional example.

The dot-product formulation is presented as the version that scales naturally to higher-dimensional inputs.

1. Audio
   Observation

   From 01:58.000 to 02:58.000 the speaker defines the directional derivative through an infinitesimal nudge in the direction of a vector.

1. Directional derivative concept
2. From coordinate partial derivatives to a directional derivative

1. Audio
   Observation

   From 00:53.000 to 01:33.000 partial derivatives are explained as x- and y-direction nudges.

2. Audio
   Observation

   From 01:33.000 to 02:58.000 the same idea is extended to an arbitrary vector direction.

1. Partial derivative as a directional nudge along coordinate axes
2. Directional derivative concept
3. s0-rel-partial-to-directional

1. Audio
   Observation

   From 02:02.000 to 02:51.000 the speaker says the step should be very small and formally uses the limit as h goes to zero.

2. Formula
   Observation

   At 02:35.000 the board writes $h\vec{v}$.

1. Directional derivative concept
2. Confusing the direction vector with the actual infinitesimal displacement

1. Formula
   Observation

   At 00:53.000 the board shows ∂f/∂$x(1,2)$.

2. Diagram
   Observation

   At 01:07.000 a horizontal arrow from (1,2) illustrates the x-direction nudge.

1. Partial derivative as a directional nudge along coordinate axes
2. Horizontal partial-derivative visualization

1. Formula
   Observation

   At 01:33.000 the board writes $\vec{v} = [-1,2]$.

2. Diagram
   Observation

   At 01:51.000 a purple arrow from (1,2) shows that direction.

1. Introducing an arbitrary direction vector
2. Slanted direction vector from the base point

1. Audio
   Observation

   how you end up calculating it

1. Calculating Directional Derivative from Components
2. General Formula for Directional Derivative

1. Audio
   Observation

   written... with respect to the gradient... looks like a dot product

1. Directional Derivative as Dot Product with Gradient
2. ∇f

1. Audio
   Observation

   whole bunch of other notations too

1. Notation for Directional Derivative

1. Formula
   Observation

   Red box formula $\nabla_{\vec w} f = a\frac{\partial f}{\partial x} + b\frac{\partial f}{\partial y}$.

1. Directional derivative in a general vector direction

1. Audio
   Observation

   Speaker identifies the partial-derivative vector as the gradient and writes $\vec w\cdot\nabla f$.

1. Directional derivative as a dot product with the gradient
2. Gradient vector of a two-variable function

1. Audio
   Observation

   Speaker contrasts $\nabla_{\vec w} f$ with $\nabla f$.

1. Directional derivative in a general vector direction
2. Gradient vector of a two-variable function

1. Formula
   Observation

   Yellow formula for $\vec v=(-1,2)$.

1. Directional derivative in the specific direction $\vec v=(-1,2)$