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

Hey there, future calculus masters! 👋 Let's make sure you're totally prepped for the AP exam. This guide is designed to be your best friend the night before the test—clear, concise, and super helpful. We're focusing on derivative rules today, so let's jump right in!

2.6 Derivative Rules: Constant, Sum, Difference, and Constant Multiple

These rules are your bread and butter for quickly finding derivatives of polynomial functions. Let's break them down:

• What it is: The derivative of any constant is always zero.
• Formula: If $f(x) = c$ (where $c$ is a constant), then $f'(x) = 0$.
• Example: If $f(x) = 7$, then $f'(x) = 0$.

• What it is: The derivative of a sum of functions is the sum of their derivatives.
• Formula: If $f(x) = g(x) + h(x)$, then $f'(x) = g'(x) + h'(x)$.
• Example: If $f(x) = x^2 + 3x$, then $f'(x) = 2x + 3$.

• What it is: The derivative of a difference of functions is the difference of their derivatives.
• Formula: If $f(x) = g(x) - h(x)$, then $f'(x) = g'(x) - h'(x)$.
• Example: If $f(x) = 4x^3 - 2x$, then $f'(x) = 12x^2 - 2$.

• What it is: The derivative of a constant times a function is the constant times the derivative of the function.
• Formula: If $f(x) = c \cdot g(x)$, then $f'(x) = c \cdot g'(x)$.
• Example: If $f(x) = 5x^4$, then $f'(x) = 5 \cdot 4x^3 = 20x^3$.

🏋️‍♂️ Practice Problems: Let's Get These Rules Down!

Example 1

Consider the function $f(x) = 2x^2 + 2$. Find the derivative.

• Step 1: Identify the functions: $g(x) = 2x^2$ and $h(x) = 2$.
• Step 2: Take the derivatives: $g'(x) = 4x$ and $h'(x) = 0$.
• Result: $f'(x) = 4x + 0 = 4x$.

Example 2

Consider the function $f(x) = 100$. Find the derivative.

• Rule: Constant Rule
• Result: $f'(x) = 0$.

Example 3

Consider the function $f(x) = 5(5x+10)$. Find the derivative.

• Step 1: Identify the constant and function: $c = 5$ and $g(x) = 5x + 10$.
• Step 2: Take the derivative of $g(x)$: $g'(x) = 5$.
• Result: $f'(x) = 5 \cdot 5 = 25$.

Example 4

Consider the function $f(x) = 3x^3 - 6x$. Find the derivative.

• Step 1: Identify the functions: $g(x) = 3x^3$ and $h(x) = 6x$.
• Step 2: Take the derivatives: $g'(x) = 9x^2$ and $h'(x) = 6$.
• Result: $f'(x) = 9x^2 - 6$.

Practice Question

Multiple Choice Questions

1. If $f(x) = 4x^3 - 6x^2 + 2x - 8$, then $f'(x)$ is: (A) 12x^2 - 12x + 2 (B) 12x^2 - 12x + 2x (C) 12x^2 - 12x + 2 - 8 (D) 4x^2 - 6x + 2

2. The derivative of $g(x) = 7x^5 + 3$ is: (A) 35x^4 (B) 35x^4 + 3 (C) 7x^4 (D) 7x^5

Free Response Question

Let $h(x) = 2x^4 - 5x^3 + 7x - 10$.

(a) Find $h'(x)$. (b) Find $h''(x)$.

Answer Key

Multiple Choice

1. (A)
2. (A)

Free Response

(a) $h'(x) = 8x^3 - 15x^2 + 7$ (1 point for each correct term) (b) $h''(x) = 24x^2 - 30x$ (1 point for each correct term)

🤔 Combining the Power Rule with Other Derivative Rules

Now it's time to mix these rules with the power rule. Remember, the AP exam loves to combine concepts, so this is crucial!

Steps to Success

1. Differentiate each term using the power rule.
2. Combine the derivatives using the sum or difference rule as needed.

Example 1: Polynomial Function

Let's find $f'(x)$ for $f(x) = 3x^4 - 2x^3 + 5x^2 - 7x + 9$.

• Step 1: Differentiate each term:
  • $\frac{d}{dx}(3x^4) = 12x^3$
  • $\frac{d}{dx}(-2x^3) = -6x^2$
  • $\frac{d}{dx}(5x^2) = 10x$
  • $\frac{d}{dx}(-7x) = -7$
  • $\frac{d}{dx}(9) = 0$
• Step 2: Combine the derivatives: $f'(x) = 12x^3 - 6x^2 + 10x - 7$.

Example 2: Polynomial with Constant Multiples

Let's find $g'(x)$ for $g(x) = 2x^5 - 3x^4 + 6x^3$.

• Step 1: Differentiate each term:
  • $\frac{d}{dx}(2x^5) = 10x^4$
  • $\frac{d}{dx}(-3x^4) = -12x^3$
  • $\frac{d}{dx}(6x^3) = 18x^2$
• Step 2: Combine the derivatives: $g'(x) = 10x^4 - 12x^3 + 18x^2$.

Final Exam Focus 🎯

• High-Priority Topics: These derivative rules are foundational. Master them!
• Common Question Types: Expect to see these rules in combination with the power rule and other rules we'll cover later.
• Time Management: Practice makes perfect. Work through problems quickly and accurately.
• Common Pitfalls: Watch out for simple arithmetic errors and missed negative signs.

Alright, you've got this! Keep practicing, stay confident, and you'll rock the AP Calculus exam. Let's get that 5! 💪