Derivatives of cos x, sinx, e^x, and ln x

Benjamin Wright
6 min read
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Study Guide Overview
This study guide covers derivatives of special functions, including sin(x), cos(x), eหฃ, and ln(x). It provides a quick reference table of these derivatives, examples of how to apply them, common mistakes to avoid, and practice questions. The guide emphasizes memorizing the derivatives, combining them with other derivative rules (like the chain rule), and applying them in the context of AP exam questions.
Derivatives of Special Functions: Your Cheat Sheet ๐
Hey there! Let's dive into the derivatives of those special functions you'll see all the time: , , , and . The good news? Once you memorize these rules, they're super straightforward! This guide is designed to make sure you're ready to nail these on the AP exam. Let's get started!
Quick Reference Table
Here's a handy table summarizing the derivatives of these special functions. Keep this close โ it's your secret weapon! ๐
Function | Derivative |
---|---|
Memorize these! Theyโre the building blocks for more complex problems. Use the mnemonic "Sine goes to Cosine" (and remember the negative for cosine) to help you remember.
Derivative of
The derivative of is always . Simple as that! Let's see it in action:
Example:
To find , we take the derivative of each term separately:
- The derivative of
4\sin x
is4\cos x
(since the derivative of is ). - The derivative of
3x
is3
(using the power rule).
So, .
Remember to apply the constant multiple rule when a function is multiplied by a constant (like the 4 in 4\sin x
).
Derivative of
The derivative of is always . Don't forget that negative sign! It's a common place to slip up. ๐ฌ
Example:
Let's break it down:
- The derivative of
2\cos x
is (since the derivative of is ). - The derivative of the constant
3
is0
.
Therefore, .
Forgetting the negative sign when differentiating is a very common error. Double-check your signs!
Derivative of
This is the easiest one! The derivative of is just... . Yes, it stays the same! ๐คฏ
Example:
Let's find the derivative:
- The derivative of is .
- The derivative of
3x^4
is12x^3
(using the power rule).
So, .
is the only function that is its own derivative. It's special!
Derivative of
The derivative of is . Got it? ๐
Example:
Let's find :
- The derivative of
5\ln x
is (since the derivative of is ). - The derivative of
2x
is2
.
Thus, .
Think of "ln" as "one over," then put the x in the denominator. Derivative of is .
Final Exam Focus
Okay, let's talk strategy for the exam. Hereโs what you need to focus on:
- Memorize the Basic Derivatives: The derivatives of , , , and are crucial. Know them inside and out.
- Combine Rules: AP questions often mix these with other derivative rules (power rule, product rule, quotient rule, chain rule). Practice combining them!
- Watch for Trig Identities: Sometimes, you might need to simplify using trig identities before taking a derivative.
- Chain Rule: Don't forget the chain rule when the argument of these functions is not just (e.g., or ).
Last-Minute Tips:
- Time Management: Don't spend too long on any one question. If you're stuck, move on and come back later.
- Show Your Work: Even if you don't get the final answer, you can earn partial credit for showing your steps.
- Check Your Signs: Double-check for negative signs, especially with derivatives.
- Practice, Practice, Practice: The more you practice, the more confident you'll feel. Do as many practice problems as you can.
These derivatives are fundamental and appear in many contexts. Mastering them will significantly boost your score.
Practice Questions
Let's put your knowledge to the test!
Practice Question
Multiple Choice Questions
-
What is the derivative of ? (A)
3\cos x - 2e^x
(B) (C)3\cos x + 2e^x
(D) -
If , what is ? (A)
4\sin x + \frac{1}{x}
(B) (C)4\sin x - \frac{1}{x}
(D) -
Find the derivative of ? (A)
5e^x - 2\cos x + 7
(B)5e^x + 2\cos x
(C)5e^x - 2\cos x
(D)5e^x + 2\cos x + 7
Free Response Question
Consider the function
(a) Find . (2 points) (b) Find the slope of the tangent line to the graph of at . (2 points) (c) Find the equation of the tangent line to the graph of at . (3 points) (d) Find . (2 points)
Answer Key
Multiple Choice Answers
- (A)
3\cos x - 2e^x
- (B)
- (C)
5e^x - 2\cos x
Free Response Question
(a) (2 points - 1 point for each correct derivative) (b) (2 points - 1 point for correct substitution, 1 point for correct answer) (c) . The tangent line is or (3 points - 1 point for correct , 1 point for correct slope, 1 point for correct equation) (d) (2 points - 1 point for each correct derivative)
You've got this! Keep practicing, and you'll be ready to ace the exam. Good luck! ๐

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