#The Product Rule 🚀

Welcome back! Let's master the product rule, a crucial tool for finding derivatives of functions multiplied together. This is a key skill that you'll see throughout the AP Calculus exam.

#🏁 Product Rule Definition

The product rule helps us find the derivative of two functions multiplied together. It's not as simple as just multiplying the derivatives!

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

#✏️ Product Rule: Walkthrough

Let's break it down with an example:

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

Applying the product rule:

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

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

Here's a visual to show the difference between the correct and incorrect derivative:

#🧮 Product Rule: Practice Problems

Let's solidify your understanding with some practice!

#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

First, expand the function:

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

Now, take the derivative:

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

#Solving Example 1 With Product Rule

Apply the product rule:

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

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

#Product Rule: Example 2

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

Using the product rule:

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

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

#Product Rule: Example 3

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

Remember that the derivative of $e^x$ is $e^x$!

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

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

#🌟 Closing

Great job! You've now got a solid grasp of the product rule. This is a fundamental concept, so make sure you're comfortable with it. You'll definitely see it on the AP exam.

The product rule is a high-value topic. Expect to see it in multiple-choice and free-response questions.

#Final Exam Focus

• Master the Basics: The product rule is a foundational concept. Make sure you can apply it quickly and accurately.
• Look for Opportunities: Be on the lookout for functions that are products.
• Don't Simplify Unless Necessary: Save time by not simplifying unless the question explicitly asks you to.
• Combine with Other Rules: The product rule often appears with other derivative rules (chain rule, etc.).

#Practice Questions

Practice Question

#Multiple Choice Questions

1. If $f(x) = x^2 \sin(x)$, what is $f'(x)$? (A) $2x \cos(x)$ (B) $2x \sin(x) + x^2 \cos(x)$ (C) $2x \sin(x) - x^2 \cos(x)$ (D) $x^2 \cos(x)$

2. Find the derivative of $y = (x^3 + 1)e^x$ (A) $3x^2e^x$ (B) $(x^3 + 1)e^x + 3x^2$ (C) $(x^3 + 3x^2 + 1)e^x$ (D) $3x^2e^x + (x^3 + 1)e^x$

#Free Response Question

Let $f(x) = x^2 \cos(x)$.

(a) Find $f'(x)$.

(b) Find the equation of the tangent line to the graph of $f$ at $x = \pi$.

(c) Find all values of x for which $f'(x) = 0$ on the interval $[0, 2\pi]$.

Scoring Rubric:

(a) 2 points * 1 point for correct application of the product rule * 1 point for the correct derivative

(b) 3 points * 1 point for finding the correct slope at $x=\pi$ * 1 point for correct $y$-value at $x=\pi$ * 1 point for the equation of the tangent line

(c) 4 points * 1 point for setting $f'(x) = 0$ * 1 point for factoring or simplifying * 2 points for correct x-values

#Answers

Multiple Choice:

1. (B)
2. (C)

Free Response:

(a) $f'(x) = 2x \cos(x) - x^2 \sin(x)$

(b) $f(\pi) = \pi^2 \cos(\pi) = -\pi^2$. $f'(\pi) = 2\pi \cos(\pi) - \pi^2 \sin(\pi) = -2\pi$. The tangent line is $y - (-\pi^2) = -2\pi(x - \pi)$ or $y = -2\pi x + \pi^2$.

(c) $f'(x) = x(2\cos(x) - x\sin(x)) = 0$. $x = 0$ or $2\cos(x) - x\sin(x) = 0$. Using a calculator, the approximate values are $x=0, 0.860, 2.289, 4.273$.