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

Abigail Young
6 min read
Listen to this study note
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 ๐
Hey there, future calculus masters! ๐ 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 testโclear, 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:
The Constant Rule
- What it is: The derivative of any constant is always zero.
- Formula: If (where is a constant), then .
- Example: If , then .
The Sum Rule
- What it is: The derivative of a sum of functions is the sum of their derivatives.
- Formula: If , then .
- Example: If , then .
The Difference Rule
- What it is: The derivative of a difference of functions is the difference of their derivatives.
- Formula: If , then .
- Example: If , then .
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 , then .
- Example: If , then .
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.
๐๏ธโโ๏ธ Practice Problems: Let's Get These Rules Down!
Example 1
Consider the function . Find the derivative.
- Step 1: Identify the functions: and .
- Step 2: Take the derivatives: and .
- Result: .
Example 2
Consider the function . Find the derivative.
- Rule: Constant Rule
- Result: .
Example 3
Consider the function . Find the derivative.
- Step 1: Identify the constant and function: and .
- Step 2: Take the derivative of : .
- Result: .
Example 4
Consider the function . Find the derivative.
- Step 1: Identify the functions: and .
- Step 2: Take the derivatives: and .
- Result: .
Practice Question
Multiple Choice Questions
-
If , then is: (A)
12x^2 - 12x + 2
(B)12x^2 - 12x + 2x
(C)12x^2 - 12x + 2 - 8
(D)4x^2 - 6x + 2
-
The derivative of is: (A)
35x^4
(B)35x^4 + 3
(C)7x^4
(D)7x^5
Free Response Question
Let .
(a) Find . (b) Find .
Answer Key
Multiple Choice
- (A)
- (A)
Free Response
(a) (1 point for each correct term) (b) (1 point for each correct term)
๐ค 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
- Differentiate each term using the power rule.
- Combine the derivatives using the sum or difference rule as needed.
Example 1: Polynomial Function
Let's find for .
- Step 1: Differentiate each term:
- Step 2: Combine the derivatives: .
Example 2: Polynomial with Constant Multiples
Let's find for .
- Step 1: Differentiate each term:
- Step 2: Combine the derivatives: .
Pro Tip: Always double-check your exponents and coefficients. A small mistake can throw off the entire problem!
Final Exam Focus ๐ฏ
- 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.
Watch out! Don't forget the constant rule! Constants alone become zero when differentiated.
The derivative of is always 1. This is a quick fact that can save time on the exam.
Power Rule Reminder: "Bring down the power, reduce by one!" This simple phrase will help you remember how to apply the power rule correctly.
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! ๐ช

How are we doing?
Give us your feedback and let us know how we can improve
Question 1 of 9
What is the derivative of ?
15
1
0
x