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Calculus · English

Product rule | Derivative rules | AP Calculus AB | Khan Academy

Learn the derivative product rule and complete the example x² sin x, with bilingual notes, differentiability conditions and radians clarified.

Reviewed learning material · Video analysis · English

Apply the product rule to differentiate a product of two functions: differentiate one factor, keep the other, then add the complementary term. Khan Academy demonstrates the complete example x² sin x, obtaining 2x sin x+x² cos x with consistent colors for the two factors. The lesson teaches application rather than a proof of the general rule. Editorial notes state the differentiability conditions and the use of radians for trigonometric derivatives.

Before you watch

Basic understanding of what a derivative is.
Knowledge of how to differentiate simple power functions (like x^n) and trigonometric functions (like sin(x)).
Power Rule
Derivatives of Trigonometric Functions

Chapters

0:00Introduction to the Product Rule 0:13The Product Rule Formula 0:58Applying the Product Rule: An Example 1:20Continue the worked example 1:30Identifying f(x) and g(x)1:50Finding Individual Derivatives 2:08Applying the Product Rule 2:30Final Simplified Answer

Learning script

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

Introduce the product rule as a tool for differentiating products. This lesson will apply the rule without proving the general theorem.

When f and g are differentiable at the point, differentiate one factor at a time: f'g+fg'. Keep the other factor unchanged in each term, and then add the terms.

Set up the example x² sin x as a product of the squared function and sine; the following part of the same video completes the calculation.

Continue differentiating $x^2 \sin x$ , using $\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$ .

Choose $f(x) = x^2$ in pink and $g(x) = \sin x$ in green.

The power rule gives $f'(x) = 2x$ , and, with x in radians, the sine derivative gives $g'(x) = \cos x$ .

Substitute the two pairs: $f'(x)g(x)$ becomes $(2x)(\sin x)$ , and $f(x)g'(x)$ becomes $(x^2)(\cos x)$ .

The final result is $2x \sin x + x^2 \cos x$ . The two colors track where the original factors and derivatives appear.

Knowledge cards

01

Product Rule Formula

For f and g differentiable at the point, differentiate one factor at a time while retaining the other, then add the two terms.

\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)

02

Example Setup: Derivative of x^2 sin(x)

Split x² sin x into f(x)=x² and g(x)=sin x. This setup interval identifies the factors; the full video then computes both derivatives and the final answer.

03

Product Rule Formula

The product rule applies when both factors are differentiable at the point; its two summands keep the undifferentiated partner factor.

\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)

04

Example: Derivative of x^2 sin(x)

Let $f(x) = x^2$ and $g(x) = \sin x$ . Then $f'(x) = 2x$ and, using radians, $g'(x) = \cos x$ . The rule gives $(2x)(\sin x) + (x^2)(\cos x)$ , or $2x \sin x + x^2 \cos x$ .

\frac{d}{dx}[x^2 \sin x] = 2x \sin x + x^2 \cos x

05

Derivative of Sine

With the argument measured in radians, the derivative of sine is cosine.

\frac{d}{dx}(\sin x) = \cos x

Detailed learning notes

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

Symbols · 13

f(x)

Clear evidence

Shown in the video

Evidence

Formula
Observation
f(x) is written in pink.
Audio
Observation
Narration identifies f as the first factor.

Symbol

f(x)

Meaning

The first function in the product.

Domain

Unspecified in the video.

g(x)

Clear evidence

Shown in the video

Evidence

Formula
Observation
g(x) is written in green.
Audio
Observation
Narration identifies g as the second factor.

Symbol

g(x)

Meaning

The second function in the product.

Domain

Unspecified in the video.

d/dx

Clear evidence

Shown in the video

Evidence

Formula
Observation
d/dx is written in white.
Audio
Observation
The presenter introduces the operation of differentiating the product.

Symbol

d/dx

Meaning

The derivative operator with respect to x.

Domain

Applied to functions of x.

f'(x)

Clear evidence

Shown in the video

Evidence

Formula
Observation
f'(x) is written in pink.
Audio
Observation
The presenter differentiates the first factor.

Symbol

f'(x)

Meaning

The derivative of f(x) with respect to x.

Domain

Unspecified in the video.

g'(x)

Clear evidence

Shown in the video

Evidence

Formula
Observation
g'(x) is written in green.
Audio
Observation
The presenter differentiates the second factor.

Symbol

g'(x)

Meaning

The derivative of g(x) with respect to x.

Domain

Unspecified in the video.

x^2

Clear evidence

Shown in the video

Evidence

Formula
Observation
x^2 is written in white.
Audio
Observation
The first example factor is introduced as x squared.

Symbol

x^2

Meaning

A specific function used in an example.

Domain

Unspecified in the video.

\sin x

Clear evidence

Shown in the video

Evidence

Formula
Observation
sin x is written in white.
Audio
Observation
The other example factor is introduced as the sine function.

Symbol

\sin x

Meaning

A specific trigonometric function used in an example.

Domain

Unspecified in the video.

\frac{d}{dx}

Clear evidence

Shown in the video

Evidence

Formula
Observation
The operator $\frac{d}{dx}$ is written in yellow at the top and used for the example.

Symbol

\frac{d}{dx}

Meaning

Derivative with respect to x

Domain

Calculus

f(x)

Clear evidence

Shown in the video

Evidence

Formula
Observation
$f(x)$ appears in the product rule formula and is assigned to $x^2$ .

Symbol

f(x)

Meaning

First function in the product rule; set to $x^2$ in the example

Domain

Real-valued functions

g(x)

Clear evidence

Shown in the video

Evidence

Formula
Observation
$g(x)$ appears in the product rule formula and is assigned to $\sin x$ .

Symbol

g(x)

Meaning

Second function in the product rule; set to $\sin x$ in the example

Domain

Real-valued functions

f'(x)

Clear evidence

Shown in the video

Evidence

Formula
Observation
$f'(x)$ appears in the product rule formula and is calculated as $2x$ .

Symbol

f'(x)

Meaning

Derivative of $f(x)$

Domain

Real-valued functions

g'(x)

Clear evidence

Shown in the video

Evidence

Formula
Observation
$g'(x)$ appears in the product rule formula and is calculated as $\cos x$ .

Symbol

g'(x)

Meaning

Derivative of $g(x)$

Domain

Real-valued functions

Knowledge points · 5

Product Rule Formula

Clear evidence

Supplementary explanation

Evidence

Formula
Observation
The formula d/dx[f(x)g(x)] = f'(x)g(x) + f(x)g'(x) is written on the screen.
Audio
Observation
The narration explains the two additive terms by differentiating one factor at a time.

Formula

Explanation

The product rule states that the derivative of a product of two functions is equal to the derivative of the first function multiplied by the second function, plus the first function multiplied by the derivative of the second function.

Formula

\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)

Conditions

Both f and g are differentiable at the point where the derivative is taken.

Derivative Notation

Clear evidence

Shown in the video

Evidence

Formula
Observation
The notation d/dx is used.
Audio
Observation
Narration introduces differentiation with respect to the variable.

Definition

Explanation

The notation d/dx represents the operation of taking the derivative of a function with respect to the variable x.

Formula

\frac{d}{dx}

Product Rule Formula

Clear evidence

Supplementary explanation

Evidence

Formula
Observation
The general product rule formula is written at the top of the screen.
Audio
Observation
The presenter proceeds from the general rule to the worked example.

Formula

Explanation

The derivative of a product of two functions is the derivative of the first times the second plus the first times the derivative of the second.

Formula

\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)

Conditions

Both f and g are differentiable at the point where the derivative is taken.

Power Rule

Clear evidence

Supplementary explanation

Evidence

Audio
Observation
The power rule is used to differentiate the squared factor.
Formula
Observation
$f'(x) = 2x$ is written.

Method

Explanation

Used to find the derivative of $x^n$ , resulting in $nx^{n-1}$ . Applied here to find the derivative of $x^2$ .

Formula

\frac{d}{dx}(x^n) = nx^{n-1}

Conditions

For arbitrary real n, the displayed power rule applies on x>0.
In this example n=2, so the polynomial derivative is valid for every real x.

Derivative of Sine Function

Clear evidence

Supplementary explanation

Evidence

Audio
Observation
The presenter uses cosine for the derivative of sine.
Formula
Observation
$g'(x) = \cos x$ is written.

Formula

Explanation

The derivative of the sine function with respect to x is the cosine function.

Formula

\frac{d}{dx}(\sin x) = \cos x

Conditions

The argument x is measured in radians.

Claims and conditions · 1

Scope of the Video

Clear evidence

Shown in the video

Evidence

Audio
Observation
The presenter states the lesson will demonstrate use of the rule rather than prove it.

Proposition

Statement

The video will not provide a proof for the product rule, but will focus on its application.

Quantifiers

None

Derivations and proofs · 1

Worked derivation for the derivative of x² sin x

Clear evidence

Supplementary explanation

Evidence

Audio
Observation
The narration follows the substitution of both factors and both derivatives into the rule.
Formula
Observation
Step-by-step writing of the solution on the blackboard.

Intuitive argument

Steps

Expression
$\frac{d}{dx}[x^2 \sin x]$
Explanation
Identify the problem as finding the derivative of a product.
Justification
Given expression.
Shown in the video
Expression
$f(x) = x^2, g(x) = \sin x$
Explanation
Assign the individual functions to match the product rule structure.
Justification
Definition of product rule components.
Shown in the video
Expression
$f'(x) = 2x$
Explanation
Calculate the derivative of the first function.
Justification
Power rule.
Shown in the video
Expression
$g'(x) = \cos x$
Explanation
Calculate the derivative of the second function.
Justification
The power rule gives the squared-factor derivative; the sine derivative uses radians.
Supplementary explanation
Expression
$f'(x)g(x) + f(x)g'(x)$
Explanation
Write out the product rule formula with the specific functions.
Justification
Product rule formula.
Shown in the video
Expression
$(2x)(\sin x) + (x^2)(\cos x)$
Explanation
Substitute the calculated derivatives and original functions into the formula.
Justification
Algebraic substitution.
Shown in the video
Expression
$2x \sin x + x^2 \cos x$
Explanation
Simplify the expression by removing parentheses.
Justification
Algebraic simplification.
Shown in the video

Conclusion

The derivative of $x^2 \sin x$ is $2x \sin x + x^2 \cos x$ . The video concludes with this final expression.

Worked examples · 2

Setting up a Product Rule Example

Clear evidence

Shown in the video

Evidence

Formula
Observation
The expression x^2 \sin x is written on the screen.
Audio
Observation
The presenter chooses a squared factor multiplied by sine as the example.

Uncertainties

This 63–79-second setup interval precedes the worked calculation, which is completed later in the same full video.

Problem

Find the derivative of the function x^2 \sin x using the product rule.

Given

The function is f(x) = x^2 \sin x.

Goal

Set up the application of the product rule to find the derivative.

Steps

Expression
$x^2 \sin x$
Explanation
Identify the function as a product of two terms, x^2 and \sin x.
Justification
Observation from the video.
Shown in the video

Answer

This interval selects the two factors; the full continuation computes the derivative.

Verification

The selected factors multiply to the displayed example function.

Applying Product Rule to x^2 sin x

Clear evidence

Supplementary explanation

Evidence

Audio
Observation
Narration completes the example through substitution and simplification.
Formula
Observation
All mathematical steps are written on the screen.

Problem

Find the derivative of $x^2 \sin x$ .

Given

Function to differentiate: $x^2 \sin x$
Product rule formula
Both f and g are differentiable at the point where the derivative is taken.
Use radians for the sine argument.

Goal

Compute $\frac{d}{dx}[x^2 \sin x]$ .

Steps

Expression
$f(x) = x^2, g(x) = \sin x$
Explanation
Decompose the product into two functions.
Justification
Structure of the product rule.
Shown in the video
Expression
$f'(x) = 2x, g'(x) = \cos x$
Explanation
Differentiate each function separately.
Justification
The power rule gives the squared-factor derivative; the sine derivative uses radians.
Supplementary explanation
Expression
$\frac{d}{dx}[x^2 \sin x] = (2x)(\sin x) + (x^2)(\cos x)$
Explanation
Apply the product rule formula.
Justification
Product rule.
Shown in the video
Expression
$= 2x \sin x + x^2 \cos x$
Explanation
Simplify the result.
Justification
Algebra.
Shown in the video

Answer

2x \sin x + x^2 \cos x

Verification

The result matches the final expression written on the board at the end of the clip.

Visual events · 2

Writing the Product Rule Formula

Clear evidence

Shown in the video

Evidence

Animation
Observation
The formula for the product rule is written out step-by-step on a black background. Different parts of the formula are color-coded (pink for f(x) and f'(x), green for g(x) and g'(x), white for operators and other symbols).

Objects

Text 'Product Rule'
Formula d/dx[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)

Changes

The formula is progressively written on the screen.

Invariants

The black background remains constant.

Interpretation

The visual presentation breaks down the components of the product rule formula, using color coding to distinguish between the two functions and their derivatives.

Color-Coded Variable Tracking

Clear evidence

Shown in the video

Evidence

Diagram
Observation
Functions and their derivatives are color-coded (pink for f, green for g) throughout the explanation.

Objects

f(x) and its derivative
g(x) and its derivative

Changes

Colors are consistently applied to identify which parts of the formula correspond to which functions.

Invariants

The color association remains constant from definition to final substitution.

Interpretation

Visual aid to help students track the components of the product rule during substitution.

Concept relations · 3

Derivative Notation → Product Rule Formula

Clear evidence

Supplementary explanation

Evidence

Audio
Observation
The explanation uses derivative notation throughout.

Prerequisite

Explanation

Understanding the basic concept of a derivative is a prerequisite for understanding the product rule.

Power Rule → Worked derivation for the derivative of x² sin x

Clear evidence

Derived from the video

Evidence

Audio
Observation
The presenter invokes the power rule to obtain the first factor derivative.

Application

Explanation

The power rule is a prerequisite method used within the larger derivation.

Derivative of Sine Function → Worked derivation for the derivative of x² sin x

Clear evidence

Derived from the video

Evidence

Audio
Observation
The presenter uses the standard sine derivative for the other factor.

Application

Explanation

The standard derivative of sine is a prerequisite fact used within the larger derivation.

Find an answer · 4

What is the product rule for derivatives?

Clear evidence

Shown in the video

Evidence

Audio
Observation
The presenter introduces the product rule.

Knowledge points

Product Rule Formula

How do you apply the product rule to a function like x^2 sin(x)?

Clear evidence

Shown in the video

Evidence

Audio
Observation
The presenter moves from the rule to an example.

Knowledge points

Product Rule Formula
Setting up a Product Rule Example

How do you apply the product rule to find the derivative of a function like x^2 sin(x)?

Clear evidence

Shown in the video

Evidence

Audio
Observation
The example demonstrates the complete application of the rule.

Knowledge points

Product Rule Formula
Worked derivation for the derivative of x² sin x
Applying Product Rule to x^2 sin x

What is the derivative of sin(x)?

Clear evidence

Shown in the video

Evidence

Formula
Observation
g'(x) = cos x is written.

Knowledge points

Derivative of Sine Function

Coverage and review notes

Covered · Introduction to the topic and scope of the video.

Covered · Statement and explanation of the product rule formula.

Covered · Setting up an example problem to apply the product rule.

Covered · The entire clip covers the statement of the product rule and its step-by-step application to a specific example.

Explore the knowledge in this video

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Derivatives Explanation
Why this connection?
Apply the product rule to differentiate a product of two functions: differentiate one factor, keep the other, then add the complementary term. Khan Academy demonstrates the complete example x² sin x, obtaining 2x sin x+x² cos x with consistent colors for the two factors. The lesson teaches application rather than a proof of the general rule. Editorial notes state the differentiability conditions and the use of radians for trigonometric derivatives.