2.8 The Product Rule

2.8 The Product Rule

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.

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

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:

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

Using the product rule, we can find that:

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

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

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

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

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

Graph of $f(x)$

Graph of $f(x)$ created with Desmos

Graph of $f'(x)$

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

Incorrect Graph of $f'(x)$

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

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🧮 Product Rule: Practice Problems

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

Product Rule: Example 1

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

Solving Example 1 Without Product Rule

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

$$y = 6x^3 - 3x^2- 8x^2 + 4x$$ $$y = 6x^3 -11x^2 + 4x$$

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

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

Solving Example 1 With Product Rule

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

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

Therefore,

$$y' = (3x^2-4x)(2) + (2x-1)(6x-4)$$

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

Product Rule: Example 2

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

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

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

Therefore,

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

Product Rule: Example 3

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

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

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

Therefore,

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

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🌟 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.

Encouraging GIF with animated ice cream

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