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2.8 The Product Rule

3 min readfebruary 15, 2024

Welcome back to AP Calculus with Fiveable! This topic focuses on taking the derivative of a product. We’ve worked through derivatives at a point, sum and difference rules, and trigonometric derivatives, so let's keep building our derivative skills. 🙌

🏁 Product Rule Definition

To find the derivative of a product of functions, we need to multiply the first function by the derivative of the second and add it to the second function multiplied by the derivative of the first.

ddx(f(x)g(x))=f(x)g(x)+g(x)f(x)\frac{d}{dx}(\textcolor{red}{f(x)}\textcolor{green}{g(x)})= \textcolor{red}{f(x)} \textcolor{blue}{g'(x)} + \textcolor{green}{g(x)} \textcolor{pink}{f'(x)}

A fun way to remember this rule is by saying:

“First d second (first function times the derivative of the second)

Plus second d first (second function times the derivative of the first).” 😁

This is necessary because the product of derivatives of two functions does not equal the derivative of a product of two functions.

✏️ Product Rule: Walkthrough

For example, let’s find the derivative of the following function:

f(x)=sin(x)(x2+2x)f(x) = \sin(x)(x^2 + 2x)

Using the product rule, we can find that:

f(x)=sin(x)ddx(x2+2x)+(x2+2x)ddx(sin(x))\textcolor{green}{f'(x)} = \sin(x)*\frac {d}{dx}(x^2 + 2x) + (x^2+2x)*\frac {d}{dx}(\sin(x))
f(x)=sin(x)(2x+2)+(x2+2x)cos(x)\textcolor{green}{f'(x)} = \sin(x)(2x+2) + (x^2+2x)\cos(x)

If we incorrectly attempt to calculate the derivative of f(x)f(x), it would say

f(x)=cos(x)(2x+2)\textcolor{red}{f'(x)} = cos(x)( 2x + 2)

However, sin(x)(2x+2)+(x2+2x)cos(x)cos(x)(2x+2)\textcolor{green}{sin(x)(2x+2) + (x^2+2x)cos(x)} \neq \textcolor{red}{cos(x)( 2x + 2)}.

This can be seen in the following graphs. f(x)f'(x) represents the correct derivative of f(x)f(x) because the critical points and positive and negative values match the original functions.

Screen Shot 2023-12-12 at 1.58.12 PM.png

Graph of f(x)f(x) created with Desmos

Screen Shot 2023-12-12 at 1.35.31 PM.png

Graph of f(x)f'(x) created with Desmos

Screen Shot 2023-12-12 at 1.58.24 PM.png

Incorrect Graph of f(x)f'(x); Graph created with Desmos


🧮 Product Rule: Practice Problems

Let’s work on a few questions and make sure we have the concept down!

Product Rule: Example 1

Find yy' for y=(3x24x)(2x1)y = (3x^2-4x)(2x-1) with and without the Product Rule.

Solving Example 1 Without Product Rule

To find yy' without the product rule, we have to first expand the function.

y=6x33x28x2+4xy = 6x^3 - 3x^2- 8x^2 + 4x
y=6x311x2+4xy = 6x^3 -11x^2 + 4x

Now we can take the derivative, using the derivative sum rule. Therefore,

y=18x222x+4y' = 18x^2 - 22x + 4

Solving Example 1 With Product Rule

We can quickly use the product rule to solve for yy'.

y=(3x24x)ddx(2x1)+(2x1)ddx(3x24x)y' = (3x^2-4x)* \frac{d}{dx}(2x-1) + (2x-1)* \frac{d}{dx}(3x^2-4x)

Therefore,

y=(3x24x)(2)+(2x1)(6x4)y' = (3x^2-4x)(2) + (2x-1)(6x-4)

Unless specified, you do not have to simplify for the AP Calculus Exam, so this answer is perfectly acceptable! ✅

Product Rule: Example 2

Find f(x)f'(x) if f(x)=sin(x)(3x22x+5)f(x) = \sin(x)(3x^2 - 2x + 5).

Let's use the product rule to find the derivative of f(x)f(x). Don’t forget, the derivative of sin(x)\sin(x) is cos(x)\cos(x)! Brush up on your trig derivatives with this Fiveable guide: Derivatives of cos x, sinx, e^x, and ln x.

f(x)=sin(x)ddx(3x22x+5)+(3x22x+5)ddxsin(x)f'(x) = \sin(x)*\frac{d}{dx}(3x^2-2x+5) + (3x^2-2x+5)*\frac{d}{dx}\sin(x)

Therefore,

f(x)=sin(x)(6x2)+(3x22x+5)cos(x)f'(x) = \sin(x)(6x-2) + (3x^2-2x+5)\cos(x)

Product Rule: Example 3

Find yy' if y=exsin(x)y = e^x\sin(x)

Remember that the derivative of exe^x is still exe^x! Now we can use the product rule.

y=exddx(sin(x))+sin(x)ddx(ex)y'=e^x * \frac{d}{dx}(\sin(x)) + \sin(x)*\frac{d}{dx}(e^x)

Therefore,

y=excos(x)+exsin(x)y' = e^x\cos(x) + e^x\sin(x)

🌟 Closing

Great work! 🙌 The product rule is a key foundational topic for AP Calculus. You can anticipate encountering questions involving the product rule on the exam, both in multiple-choice and as part of a free response.

https://media2.giphy.com/media/ur5T6Wuw4xK2afXVmd/giphy.gif?cid=7941fdc64i8t3xwgu78ov7fo1wvrrynfxrd4d9loorgxbu2d&ep=v1_gifs_search&rid=giphy.gif&ct=g

Image Courtesy of Giphy

Key Terms to Review (10)

Chain Rule

: The chain rule is a formula used to find the derivative of a composition of two or more functions. It states that the derivative of a composite function is equal to the derivative of the outermost function times the derivative of the innermost function.

cos x

: The cosine function, denoted as cos x, is a trigonometric function that represents the ratio of the adjacent side to the hypotenuse in a right triangle. It gives the value of the x-coordinate on the unit circle corresponding to a given angle.

e^x

: The exponential function e^x represents continuous growth or decay over time. It is defined as raising Euler's number (approximately 2.71828) to the power of x.

g'(x)

: The derivative of a function g(x) represents the rate at which the function is changing at any given point. It measures the slope of the tangent line to the graph of g(x) at that point.

Key Term: (f(x) * g(x))'

: Definition: The derivative of the product of two functions f(x) and g(x). It represents the rate at which the product of f(x) and g(x) is changing with respect to x.

ln x

: The natural logarithm function, denoted as ln x, is the inverse of the exponential function e^x. It gives us the power we need to raise Euler's number (e) to obtain x.

ln(x+2)

: The natural logarithm function ln(x+2) gives us the value y such that e raised to y equals x+2. It helps us find out what exponent we need for e (approximately 2.71828) in order to get x+2.

Product Rule

: The product rule is a formula used to find the derivative of the product of two functions. It states that the derivative of the product of two functions is equal to the derivative of the first function times the second function, plus the first function times the derivative of the second function.

sin x

: Sin x refers to the trigonometric sine function, which relates an angle in a right triangle to the ratio of the length of the side opposite that angle to the length of the hypotenuse.

sin(x^2)

: The function sin(x^2) represents the sine of the square of x. It calculates the ratio of the length of the side opposite to an angle in a right triangle to the hypotenuse, where the angle is determined by squaring x.

2.8 The Product Rule

3 min readfebruary 15, 2024

Welcome back to AP Calculus with Fiveable! This topic focuses on taking the derivative of a product. We’ve worked through derivatives at a point, sum and difference rules, and trigonometric derivatives, so let's keep building our derivative skills. 🙌

🏁 Product Rule Definition

To find the derivative of a product of functions, we need to multiply the first function by the derivative of the second and add it to the second function multiplied by the derivative of the first.

ddx(f(x)g(x))=f(x)g(x)+g(x)f(x)\frac{d}{dx}(\textcolor{red}{f(x)}\textcolor{green}{g(x)})= \textcolor{red}{f(x)} \textcolor{blue}{g'(x)} + \textcolor{green}{g(x)} \textcolor{pink}{f'(x)}

A fun way to remember this rule is by saying:

“First d second (first function times the derivative of the second)

Plus second d first (second function times the derivative of the first).” 😁

This is necessary because the product of derivatives of two functions does not equal the derivative of a product of two functions.

✏️ Product Rule: Walkthrough

For example, let’s find the derivative of the following function:

f(x)=sin(x)(x2+2x)f(x) = \sin(x)(x^2 + 2x)

Using the product rule, we can find that:

f(x)=sin(x)ddx(x2+2x)+(x2+2x)ddx(sin(x))\textcolor{green}{f'(x)} = \sin(x)*\frac {d}{dx}(x^2 + 2x) + (x^2+2x)*\frac {d}{dx}(\sin(x))
f(x)=sin(x)(2x+2)+(x2+2x)cos(x)\textcolor{green}{f'(x)} = \sin(x)(2x+2) + (x^2+2x)\cos(x)

If we incorrectly attempt to calculate the derivative of f(x)f(x), it would say

f(x)=cos(x)(2x+2)\textcolor{red}{f'(x)} = cos(x)( 2x + 2)

However, sin(x)(2x+2)+(x2+2x)cos(x)cos(x)(2x+2)\textcolor{green}{sin(x)(2x+2) + (x^2+2x)cos(x)} \neq \textcolor{red}{cos(x)( 2x + 2)}.

This can be seen in the following graphs. f(x)f'(x) represents the correct derivative of f(x)f(x) because the critical points and positive and negative values match the original functions.

Screen Shot 2023-12-12 at 1.58.12 PM.png

Graph of f(x)f(x) created with Desmos

Screen Shot 2023-12-12 at 1.35.31 PM.png

Graph of f(x)f'(x) created with Desmos

Screen Shot 2023-12-12 at 1.58.24 PM.png

Incorrect Graph of f(x)f'(x); Graph created with Desmos


🧮 Product Rule: Practice Problems

Let’s work on a few questions and make sure we have the concept down!

Product Rule: Example 1

Find yy' for y=(3x24x)(2x1)y = (3x^2-4x)(2x-1) with and without the Product Rule.

Solving Example 1 Without Product Rule

To find yy' without the product rule, we have to first expand the function.

y=6x33x28x2+4xy = 6x^3 - 3x^2- 8x^2 + 4x
y=6x311x2+4xy = 6x^3 -11x^2 + 4x

Now we can take the derivative, using the derivative sum rule. Therefore,

y=18x222x+4y' = 18x^2 - 22x + 4

Solving Example 1 With Product Rule

We can quickly use the product rule to solve for yy'.

y=(3x24x)ddx(2x1)+(2x1)ddx(3x24x)y' = (3x^2-4x)* \frac{d}{dx}(2x-1) + (2x-1)* \frac{d}{dx}(3x^2-4x)

Therefore,

y=(3x24x)(2)+(2x1)(6x4)y' = (3x^2-4x)(2) + (2x-1)(6x-4)

Unless specified, you do not have to simplify for the AP Calculus Exam, so this answer is perfectly acceptable! ✅

Product Rule: Example 2

Find f(x)f'(x) if f(x)=sin(x)(3x22x+5)f(x) = \sin(x)(3x^2 - 2x + 5).

Let's use the product rule to find the derivative of f(x)f(x). Don’t forget, the derivative of sin(x)\sin(x) is cos(x)\cos(x)! Brush up on your trig derivatives with this Fiveable guide: Derivatives of cos x, sinx, e^x, and ln x.

f(x)=sin(x)ddx(3x22x+5)+(3x22x+5)ddxsin(x)f'(x) = \sin(x)*\frac{d}{dx}(3x^2-2x+5) + (3x^2-2x+5)*\frac{d}{dx}\sin(x)

Therefore,

f(x)=sin(x)(6x2)+(3x22x+5)cos(x)f'(x) = \sin(x)(6x-2) + (3x^2-2x+5)\cos(x)

Product Rule: Example 3

Find yy' if y=exsin(x)y = e^x\sin(x)

Remember that the derivative of exe^x is still exe^x! Now we can use the product rule.

y=exddx(sin(x))+sin(x)ddx(ex)y'=e^x * \frac{d}{dx}(\sin(x)) + \sin(x)*\frac{d}{dx}(e^x)

Therefore,

y=excos(x)+exsin(x)y' = e^x\cos(x) + e^x\sin(x)

🌟 Closing

Great work! 🙌 The product rule is a key foundational topic for AP Calculus. You can anticipate encountering questions involving the product rule on the exam, both in multiple-choice and as part of a free response.

https://media2.giphy.com/media/ur5T6Wuw4xK2afXVmd/giphy.gif?cid=7941fdc64i8t3xwgu78ov7fo1wvrrynfxrd4d9loorgxbu2d&ep=v1_gifs_search&rid=giphy.gif&ct=g

Image Courtesy of Giphy

Key Terms to Review (10)

Chain Rule

: The chain rule is a formula used to find the derivative of a composition of two or more functions. It states that the derivative of a composite function is equal to the derivative of the outermost function times the derivative of the innermost function.

cos x

: The cosine function, denoted as cos x, is a trigonometric function that represents the ratio of the adjacent side to the hypotenuse in a right triangle. It gives the value of the x-coordinate on the unit circle corresponding to a given angle.

e^x

: The exponential function e^x represents continuous growth or decay over time. It is defined as raising Euler's number (approximately 2.71828) to the power of x.

g'(x)

: The derivative of a function g(x) represents the rate at which the function is changing at any given point. It measures the slope of the tangent line to the graph of g(x) at that point.

Key Term: (f(x) * g(x))'

: Definition: The derivative of the product of two functions f(x) and g(x). It represents the rate at which the product of f(x) and g(x) is changing with respect to x.

ln x

: The natural logarithm function, denoted as ln x, is the inverse of the exponential function e^x. It gives us the power we need to raise Euler's number (e) to obtain x.

ln(x+2)

: The natural logarithm function ln(x+2) gives us the value y such that e raised to y equals x+2. It helps us find out what exponent we need for e (approximately 2.71828) in order to get x+2.

Product Rule

: The product rule is a formula used to find the derivative of the product of two functions. It states that the derivative of the product of two functions is equal to the derivative of the first function times the second function, plus the first function times the derivative of the second function.

sin x

: Sin x refers to the trigonometric sine function, which relates an angle in a right triangle to the ratio of the length of the side opposite that angle to the length of the hypotenuse.

sin(x^2)

: The function sin(x^2) represents the sine of the square of x. It calculates the ratio of the length of the side opposite to an angle in a right triangle to the hypotenuse, where the angle is determined by squaring x.


© 2024 Fiveable Inc. All rights reserved.

AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.


© 2024 Fiveable Inc. All rights reserved.

AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.