### The Product Rule: AP Calculus AB/BC Study Guide

#### Introduction

Welcome to AP Calculus, where we dive deep into the exciting world of differentiation! In this guide, we'll explore the Product Rule, a powerful tool for finding derivatives of products of functions. Imagine you’re baking a cake and need to combine ingredients in the right way to get the perfect mix – the Product Rule is like your recipe for differentiating products. 🍰📚

#### The Product Rule Definition

First things first, what exactly is the Product Rule? When you need to find the derivative of a product of two functions, ( f(x) ) and ( g(x) ), the rule comes in handy. It states that:

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

Think of it as a calculus symphony where each function plays its part: one stays the same, while the other performs a derivative solo, then they switch roles. 🎶

To remember it easily, you can use this catchy phrase: “First d’ second, plus second d’ first.” 😁

This is critical (pun intended) because the derivative of a product is **not** the product of derivatives!

#### Walkthrough Example

Let’s walk through a classic example. Suppose we have:

[ f(x) = \sin(x) (x^2 + 2x) ]

Using the Product Rule, we start by taking the derivative:

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

Perform the derivatives:

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

So the derivative is:

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

High fives all around – you’ve successfully used the Product Rule! 🙌

#### Common Mistakes and Missteps

Let's see what happens if we don't use the Product Rule correctly. For instance:

[ f'(x) = \cos(x) (2x + 2) ]

This is **not** equivalent to the correct derivative, because:

[ \sin(x)(2x + 2) + (x^2 + 2x) \cos(x) \neq \cos(x) (2x + 2) ]

It's like forgetting the sugar in your cake batter – it doesn’t come out right!

#### Visualizing with Graphs

Graphs can help us see why the correct derivative works. Plotting both the function ( f(x) ) and its derivative ( f'(x) ) can show how their critical points and behaviors match up. Incorrect derivatives will lead you astray like a faulty GPS.

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) ); created with Desmos)

#### Practice Problems

Time to get our hands dirty! Let's work through a few problems to make sure you’ve got the Product Rule down pat.

**Example 1:**

Find ( y' ) for ( y = (3x^2 - 4x)(2x - 1) ).

**Without the Product Rule:**

Expand the function first:

[ y = (3x^2 - 4x)(2x - 1) ]

[ y = 3x^2 \cdot 2x + 3x^2 \cdot (-1) - 4x \cdot 2x - 4x \cdot (-1) ]

[ y = 6x^3 - 3x^2 - 8x^2 + 4x ]

Simplify:

[ y = 6x^3 - 11x^2 + 4x ]

Now differentiate:

[ y' = 18x^2 - 22x + 4 ]

**With the Product Rule:**

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

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

Simple, isn’t it?

**Example 2:**

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

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

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

Another product conquered! 🎉

**Example 3:**

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

Remember, the derivative of ( e^x ) is ( e^x ).

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

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

Practice makes perfect! 🤓

#### Key Concepts to Review

**Chain Rule:**Used to find the derivative of the composition of functions.**Cos(x):**Trigonometric function representing the adjacent/hypotenuse ratio in a right triangle or the x-coordinate on the unit circle.**( e^x ):**Exponential function representing continuous growth or decay.**( g'(x) ):**The rate of change of function ( g(x) ).**( (f(x) \cdot g(x))' ):**The derivative of the product of two functions.**Ln(x):**The natural logarithm, inverse of the exponential function.**Ln(x + 2):**The natural logarithm for the shifted argument ( x + 2 ).**Product Rule:**Swan song of differentiation for function products.**Sin(x):**The trigonometric sine function for opposite/hypotenuse ratio or y-coordinate on the unit circle.**Sin(x^2):**Sine of the square of ( x ), finding a ratio of sides based on ( x^2 ).

#### Closing

And there you have it! You’ve mastered the Product Rule for differentiation. You’ll find that this rule is your trusty sidekick throughout calculus, popping up in exams and problem sets alike.

Keep practicing, stay sharp, and remember: calculus isn't just a subject—it’s a way of life! 🧠📈

#### Key Terms to Review (10)

**Chain Rule:**The formula for differentiating composites of functions.**Cos(x):**Trigonometric function for the cosine of an angle.**( e^x ):**The exponential function, a key function in growth/decay models.**( g'(x) ):**Derivative of the function ( g(x) ), representing its rate of change.**( (f(x) \cdot g(x))' ):**Derivative of the product of ( f(x) ) and ( g(x) ).**Ln(x):**The natural logarithm, inverse of the exponential function.**Ln(x + 2):**Natural logarithm for the function shifted by 2.**Product Rule:**The differentiation rule for products of functions.**Sin(x):**The sine function, essential in trigonometry.**Sin(x^2):**Sine of the square of ( x ), another composite function.

Happy Differentiating!