#AP Calculus AB/BC: Derivative Rules - Your Ultimate Review 🚀

Hey there, future AP Calculus master! You've come a long way, and now it's time to solidify your understanding of derivative rules. This guide is designed to be your go-to resource, especially the night before the exam. Let's make sure you feel confident and ready!

# 1. Mastering Derivative Techniques

This section focuses on the core derivative rules you've learned. Remember, the AP exam loves to test your ability to recognize which rule to apply in different situations. Let's break it down:

#1.1 Basic Rules

Quick Fact

Power Rule: If $f(x) = x^n$ , then $f'(x) = nx^{n-1}$
- Example: If $f(x) = x^3$ , then $f'(x) = 3x^2$
Constant Multiple Rule: If $f(x) = cf(x)$ , then $f'(x) = cf'(x)$
- Example: If $f(x) = 5x^2$ , then $f'(x) = 10x$
Sum/Difference Rule: If $h(x) = f(x) pm g(x)$ , then $h'(x) = f'(x) pm g'(x)$
- Example: If $h(x) = x^3 + 2x$ , then $h'(x) = 3x^2 + 2$

#1.2 Trigonometric Derivatives

Quick Fact

$\frac{d}{dx} \sin(x) = \cos(x)$
$\frac{d}{dx} \cos(x) = -\sin(x)$
$\frac{d}{dx} \tan(x) = \sec^2(x)$
$\frac{d}{dx} \csc(x) = -\csc(x)\cot(x)$
$\frac{d}{dx} \sec(x) = \sec(x)\tan(x)$
$\frac{d}{dx} \cot(x) = -\csc^2(x)$

#1.3 Exponential and Logarithmic Derivatives

Quick Fact

$\frac{d}{dx} e^x = e^x$
$\frac{d}{dx} a^x = a^x \ln(a)$
$\frac{d}{dx} \ln(x) = \frac{1}{x}$
$\frac{d}{dx} \log_a(x) = \frac{1}{x\ln(a)}$
- Example: If $f(x) = e^{x}$ , then $f'(x) = e^{x}$ . If $g(x) = 2^x$ , then $g'(x) = 2^x \ln(2)$

#1.4 Product and Quotient Rules

Key Concept

Product Rule: If $h(x) = f(x)g(x)$ , then $h'(x) = f'(x)g(x) + f(x)g'(x)$
- Memory Aid: "First times the derivative of the second, plus the second times the derivative of the first."
Quotient Rule: If $h(x) = \frac{f(x)}{g(x)}$ , then $h'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}$
- Memory Aid: "Low d-high minus high d-low, over low squared."

#1.5 Chain Rule

Key Concept

Chain Rule: If $h(x) = f(g(x))$ , then $h'(x) = f'(g(x)) cdot g'(x)$
- Memory Aid: "Derivative of the outside, keep the inside, times the derivative of the inside."
- Example: If $h(x) = \sin(x^2)$ , then $h'(x) = \cos(x^2) cdot 2x$

Exam Tip

Pro Tip: Practice recognizing composite functions quickly. The chain rule is everywhere!

# 2. Implicit Differentiation

#2.1 When to Use It

Quick Fact

Use implicit differentiation when you can't easily solve for y in terms of x.
- Example: $x^2 + y^2 = 25$

#2.2 The Process

Exam Tip

Differentiate both sides of the equation with respect to x.
Remember to use the chain rule when differentiating terms involving y.
Solve for $\frac{dy}{dx}$ .
- Example: If $x^2 + y^2 = 25$ , then $2x + 2y\frac{dy}{dx} = 0$ , so $\frac{dy}{dx} = -\frac{x}{y}$

# 3. Derivatives of Inverse Functions

#3.1 Key Concept

Key Concept

If $f$ and $g$ are inverse functions, then $f(g(x)) = x$ and $g(f(x)) = x$ .

#3.2 Formula

Quick Fact

If $y = f^{-1}(x)$ , then $\frac{dy}{dx} = \frac{1}{f'(f^{-1}(x))}$ or $\frac{d}{dx}[f^{-1}(x)] = \frac{1}{f'(y)}$ .
- Example: If $f(x) = x^3$ , then $f^{-1}(x) = \sqrt[3]{x}$ , and the derivative of the inverse is $\frac{1}{3(\sqrt[3]{x})^2}$ .

# 4. Practice Problems 📝

Let's apply what we've learned! These problems are similar to what you might see on the AP exam.

#4.1 Multiple Choice Practice

Practice Question

Question 1:

Which sequence of rules can be used to differentiate $f(x) = \frac{\cos(x^{3})}{5x}$ ?

A) Quotient rule, then quotient rule again

B) Quotient rule, then chain rule

C) Chain rule, then chain rule again

D) Quotient rule, then product rule

Question 2:

Which sequence of rules can be used to differentiate $g(x) = 4x\cos(x)\sin(x)$ ?

A) Chain rule, then product rule

B) Chain rule, then chain rule again

C) Product rule, then chain rule

D) Product rule, then product rule again

Question 3:

The derivative of $h(x) = \frac{x^2}{x+1}$ is:

A) $\frac{2x}{1}$

B) $\frac{x^2 + 2x}{(x+1)^2}$

C) $\frac{x^2 + 2x}{x+1}$

D) $\frac{x^2 + 2x}{(x+1)}$

#4.2 Short Answer Practice

Practice Question

Question 4:

What is the derivative of $h(x) = (3x^{3}-15x)(2x-x^{7})$ ?

Question 5:

What is the derivative of $f(x)= 6e^{x^3+4}$ ?

#4.3 Free Response Practice

Practice Question

Question 6:

Let $f(x) = \sqrt{2x+1}$ and $g(x) = x^2 - 1$ .

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

(b) Find $g'(x)$ .

(c) Find the derivative of $h(x) = f(g(x))$ .

(d) Find the equation of the tangent line to $h(x)$ at $x = 2$ .

Solution:

(a) $f(x) = (2x+1)^{1/2}$ $f'(x) = \frac{1}{2}(2x+1)^{-1/2} \cdot 2 = \frac{1}{\sqrt{2x+1}}$ (1 point for power rule, 1 point for chain rule)

(b) $g(x) = x^2 - 1$ $g'(x) = 2x$ (1 point)

(c) $h(x) = f(g(x)) = \sqrt{2(x^2-1)+1} = \sqrt{2x^2-1}$ $h'(x) = \frac{1}{2}(2x^2-1)^{-1/2} \cdot 4x = \frac{2x}{\sqrt{2x^2-1}}$ (1 point for chain rule, 1 point for correct derivative)

(d) At $x=2$ , $h(2) = \sqrt{2(2)^2-1} = \sqrt{7}$ and $h'(2) = \frac{2(2)}{\sqrt{2(2)^2-1}} = \frac{4}{\sqrt{7}}$ The tangent line is $y - \sqrt{7} = \frac{4}{\sqrt{7}}(x-2)$ (1 point for slope, 1 point for equation)

# 5. Final Exam Focus 🎯

High-Priority Topics: Chain rule, product rule, quotient rule, implicit differentiation, and derivatives of inverse functions.
Common Question Types:
- Finding derivatives of complex functions.
- Applying derivatives in real-world contexts.
- Using implicit differentiation to find slopes.
- Solving related rates problems.
Last-Minute Tips:
- Time Management: Don't spend too long on one problem. If you're stuck, move on and come back later.
- Common Pitfalls: Watch out for sign errors and incorrect application of the chain rule.
- Strategies: Always double-check your work, especially on free-response questions. Show all steps clearly.