### Selecting Procedures for Calculating Derivatives: AP Calculus AB/BC Study Guide

#### Introduction

Welcome to the land of derivatives, where slopes reign supreme and every function has a tale to tell! 🎢 Whether you're dealing with composite, implicit, or inverse functions, mastering the art of derivatives is like unlocking the secret levels of a video game. This study guide will help you navigate the maze and come out victorious. 🏆

#### Understanding Derivative Rules

Over the past units, you've collected quite the arsenal of derivative rules like your very own toolbox. These tools range from the Power Rule, which is your basic screwdriver, to the Chain Rule, which is more like your multi-tool gadget. Let’s break down how to appropriately use these tools to tackle any derivative problem that comes your way.

Remember that knowing when to apply the right rule is key! It’s like knowing when to use the Force in a Star Wars duel. 🌟

#### Common Derivative Procedures

**Quotient Rule:** Used when you’re dealing with a function divided by another function. It’s like the Cookie-Cutter of calculus—cuts through the complexities of fractions to give you a neat result.

**Product Rule:** This one’s for when functions are getting cozy and multiplying with each other. Think of it as the "Friendship Rule"—you're dealing with products of two functions, ensuring each part gets its proper derivative credit.

**Chain Rule:** Ah, the Chain Rule—complex yet satisfying, much like solving a Rubik's Cube. Perfect for composite functions, where one function is nested inside another like a calculus Matryoshka doll.

**Implicit Differentiation:** Used for equations where y hangs out with x, and you need to find dy/dx. Think of it as detective work—solving for the hidden y’s.

#### Practice Problems and Solutions

Let’s dive into some derivative problems! Just like training at Hogwarts, practice makes perfect.

**Question 1: Differentiating [ f(x) = \frac{\cos(x^3)}{5x} ]**

**Answer:** B) Quotient rule, then chain rule

**Explanation:** The function [ f(x) = \frac{\cos(x^3)}{5x} ] involves a quotient, so we begin with the Quotient Rule. Then, we notice that (\cos(x^3)) is a composite function, requiring the Chain Rule. It’s like a two-layer cake—tackle each layer with the right tools.

**Question 2: Differentiating [ g(x) = 4x \cos(x) \sin(x) ]**

**Answer:** D) Product rule, then product rule again

**Explanation:** This fancy function [ g(x) = 4x \cos(x) \sin(x) ] is a product of three separate functions. It’s time to roll out the Product Rule twice—each application peels back another layer, like an onion. 🌰

**Question 3: What is the derivative of [ h(x) = (3x^3 - 15x)(2x - x^7) ]?**

**Solution:**

[ h(x) = (3x^3 - 15x)(2x - x^7) ]

Using the Product Rule:

[ h'(x) = (9x^2 - 15)(2x - x^7) + (3x^3 - 15x)(2 - 7x^6) ]

Simplified, the answer is:

[ h'(x) = -30x^9 + 120x^7 + 24x^3 - 60x ]

Like mixing a perfect potion, each ingredient and step must be followed precisely for the magic to work!

**Question 4: What is the derivative of [ f(x) = 6e^{x^3 + 4} ]?**

**Solution:**

[ f(x) = 6e^{x^3 + 4} ]

First, apply the Chain Rule:

[ f'(x) = 6e^{x^3 + 4} \cdot 3x^2 ]

Which simplifies to:

[ f'(x) = 18e^{x^3 + 4}x^2 ]

It’s a chain reaction! One leads to the next, resulting in a smooth derivative.

#### Key Terms to Review

**Quotient Rule:**For derivatives of two divided functions, like a divorce lawyer—splitting responsibilities.**Product Rule:**Handles the derivatives of products, ensuring both functions play nice.**Chain Rule:**Derivatives for composite functions, the ultimate nesting tool.**Implicit Differentiation:**When ( y ) is entangled with ( x ); solving for (\frac{dy}{dx}) is like untangling earbuds.

#### Conclusion

You’ve got the rules, now it’s time to play the game. Knowing which procedure to select when differentiating functions is key to mastering calculus. With practice, these rules will become second nature, like knowing how to navigate your favorite theme park without a map. 🎢

Now go forth and tackle those derivatives like a calculus ninja! 🥷 You got this! 🍀