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

Abigail Young

Abigail Young

6 min read

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Study Guide Overview

This study guide covers derivative rules for AP Calculus AB/BC, focusing on the constant, sum, difference, and constant multiple rules. It provides formulas, examples, and practice problems for each rule, including combining them with the power rule. The guide emphasizes the importance of these foundational rules for the AP exam and offers tips for success.

AP Calculus AB/BC: Derivative Rules - Your Ultimate Review ๐Ÿš€

Hey there, future calculus masters! ๐Ÿ‘‹ 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 testโ€”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:

Key Concept

The Constant Rule

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

Key Concept

The Sum Rule

  • What it is: The derivative of a sum of functions is the sum of their derivatives.
  • Formula: If f(x)=g(x)+h(x)f(x) = g(x) + h(x), then fโ€ฒ(x)=gโ€ฒ(x)+hโ€ฒ(x)f'(x) = g'(x) + h'(x).
  • Example: If f(x)=x2+3xf(x) = x^2 + 3x, then fโ€ฒ(x)=2x+3f'(x) = 2x + 3.

Key Concept

The Difference Rule

  • What it is: The derivative of a difference of functions is the difference of their derivatives.
  • Formula: If f(x)=g(x)โˆ’h(x)f(x) = g(x) - h(x), then fโ€ฒ(x)=gโ€ฒ(x)โˆ’hโ€ฒ(x)f'(x) = g'(x) - h'(x).
  • Example: If f(x)=4x3โˆ’2xf(x) = 4x^3 - 2x, then fโ€ฒ(x)=12x2โˆ’2f'(x) = 12x^2 - 2.

Key Concept

The Constant Multiple Rule

  • What it is: The derivative of a constant times a function is the constant times the derivative of the function.
  • Formula: If f(x)=cโ‹…g(x)f(x) = c \cdot g(x), then fโ€ฒ(x)=cโ‹…gโ€ฒ(x)f'(x) = c \cdot g'(x).
  • Example: If f(x)=5x4f(x) = 5x^4, then fโ€ฒ(x)=5โ‹…4x3=20x3f'(x) = 5 \cdot 4x^3 = 20x^3.
Memory Aid

Think of it like this: When you have constants hanging out with variables, they tag along for the derivative ride! But if a constant is alone, it becomes zero.

๐Ÿ‹๏ธโ€โ™‚๏ธ Practice Problems: Let's Get These Rules Down!

Example 1

Consider the function f(x)=2x2+2f(x) = 2x^2 + 2. Find the derivative.

  • Step 1: Identify the functions: g(x)=2x2g(x) = 2x^2 and h(x)=2h(x) = 2.
  • Step 2: Take the derivatives: gโ€ฒ(x)=4xg'(x) = 4x and hโ€ฒ(x)=0h'(x) = 0.
  • Result: fโ€ฒ(x)=4x+0=4xf'(x) = 4x + 0 = 4x.

Example 2

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

  • Rule: Constant Rule
  • Result: fโ€ฒ(x)=0f'(x) = 0.

Example 3

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

  • Step 1: Identify the constant and function: c=5c = 5 and g(x)=5x+10g(x) = 5x + 10.
  • Step 2: Take the derivative of g(x)g(x): gโ€ฒ(x)=5g'(x) = 5.
  • Result: fโ€ฒ(x)=5โ‹…5=25f'(x) = 5 \cdot 5 = 25.

Example 4

Consider the function f(x)=3x3โˆ’6xf(x) = 3x^3 - 6x. Find the derivative.

  • Step 1: Identify the functions: g(x)=3x3g(x) = 3x^3 and h(x)=6xh(x) = 6x.
  • Step 2: Take the derivatives: gโ€ฒ(x)=9x2g'(x) = 9x^2 and hโ€ฒ(x)=6h'(x) = 6.
  • Result: fโ€ฒ(x)=9x2โˆ’6f'(x) = 9x^2 - 6.
Practice Question

Multiple Choice Questions

  1. If f(x)=4x3โˆ’6x2+2xโˆ’8f(x) = 4x^3 - 6x^2 + 2x - 8, then fโ€ฒ(x)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)=7x5+3g(x) = 7x^5 + 3 is: (A) 35x^4 (B) 35x^4 + 3 (C) 7x^4 (D) 7x^5

Free Response Question

Let h(x)=2x4โˆ’5x3+7xโˆ’10h(x) = 2x^4 - 5x^3 + 7x - 10.

(a) Find hโ€ฒ(x)h'(x). (b) Find hโ€ฒโ€ฒ(x)h''(x).

Answer Key

Multiple Choice

  1. (A)
  2. (A)

Free Response

(a) hโ€ฒ(x)=8x3โˆ’15x2+7h'(x) = 8x^3 - 15x^2 + 7 (1 point for each correct term) (b) hโ€ฒโ€ฒ(x)=24x2โˆ’30xh''(x) = 24x^2 - 30x (1 point for each correct term)

๐Ÿค” 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 fโ€ฒ(x)f'(x) for f(x)=3x4โˆ’2x3+5x2โˆ’7x+9f(x) = 3x^4 - 2x^3 + 5x^2 - 7x + 9.

  • Step 1: Differentiate each term:
    • ddx(3x4)=12x3\frac{d}{dx}(3x^4) = 12x^3
    • ddx(โˆ’2x3)=โˆ’6x2\frac{d}{dx}(-2x^3) = -6x^2
    • ddx(5x2)=10x\frac{d}{dx}(5x^2) = 10x
    • ddx(โˆ’7x)=โˆ’7\frac{d}{dx}(-7x) = -7
    • ddx(9)=0\frac{d}{dx}(9) = 0
  • Step 2: Combine the derivatives: fโ€ฒ(x)=12x3โˆ’6x2+10xโˆ’7f'(x) = 12x^3 - 6x^2 + 10x - 7.

Example 2: Polynomial with Constant Multiples

Let's find gโ€ฒ(x)g'(x) for g(x)=2x5โˆ’3x4+6x3g(x) = 2x^5 - 3x^4 + 6x^3.

  • Step 1: Differentiate each term:
    • ddx(2x5)=10x4\frac{d}{dx}(2x^5) = 10x^4
    • ddx(โˆ’3x4)=โˆ’12x3\frac{d}{dx}(-3x^4) = -12x^3
    • ddx(6x3)=18x2\frac{d}{dx}(6x^3) = 18x^2
  • Step 2: Combine the derivatives: gโ€ฒ(x)=10x4โˆ’12x3+18x2g'(x) = 10x^4 - 12x^3 + 18x^2.
Exam Tip

Pro Tip: Always double-check your exponents and coefficients. A small mistake can throw off the entire problem!

Final Exam Focus ๐ŸŽฏ

  • 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.
Common Mistake

Watch out! Don't forget the constant rule! Constants alone become zero when differentiated.

Quick Fact

The derivative of xx is always 1. This is a quick fact that can save time on the exam.

Memory Aid

Power Rule Reminder: "Bring down the power, reduce by one!" This simple phrase will help you remember how to apply the power rule correctly.

Exam Tip

Last-Minute Tip: Do a quick run-through of your derivative rules right before the exam. It's like a mental warm-up!

Alright, you've got this! Keep practicing, stay confident, and you'll rock the AP Calculus exam. Let's get that 5! ๐Ÿ’ช

Question 1 of 9

What is the derivative of f(x)=15f(x) = 15?

15

1

0

x