zuai-logo

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

Benjamin Wright

Benjamin Wright

6 min read

Listen to this study note

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 ๐Ÿš€

Hey there! Let's dive into the derivatives of those special functions you'll see all the time: sinโกx\sin x, cosโกx\cos x, exe^x, and lnโกx\ln x. 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! ๐Ÿ˜‰

FunctionDerivative
f(x)=sinโกxf(x) = \sin xfโ€ฒ(x)=cosโกxf'(x) = \cos x
g(x)=cosโกxg(x) = \cos xgโ€ฒ(x)=โˆ’sinโกxg'(x) = -\sin x
h(x)=exh(x) = e^xhโ€ฒ(x)=exh'(x) = e^x
k(x)=lnโกxk(x) = \ln xkโ€ฒ(x)=1xk'(x) = \frac{1}{x}
Key Concept

Memorize these! Theyโ€™re 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 sinโกx\sin x

The derivative of sinโกx\sin x is always cosโกx\cos x. Simple as that! Let's see it in action:

Example:

f(x)=4sinโกx+3xf(x) = 4\sin x + 3x

To find fโ€ฒ(x)f'(x), we take the derivative of each term separately:

  • The derivative of 4\sin x is 4\cos x (since the derivative of sinโกx\sin x is cosโกx\cos x).
  • The derivative of 3x is 3 (using the power rule).

So, fโ€ฒ(x)=4cosโกx+3f'(x) = 4\cos x + 3.

Exam Tip

Remember to apply the constant multiple rule when a function is multiplied by a constant (like the 4 in 4\sin x).

Derivative of cosโกx\cos x

The derivative of cosโกx\cos x is always โˆ’sinโกx-\sin x. Don't forget that negative sign! It's a common place to slip up. ๐Ÿ˜ฌ

Example:

f(x)=2cosโกx+3f(x) = 2\cos x + 3

Let's break it down:

  • The derivative of 2\cos x is โˆ’2sinโกx-2\sin x (since the derivative of cosโกx\cos x is โˆ’sinโกx-\sin x).
  • The derivative of the constant 3 is 0.

Therefore, fโ€ฒ(x)=โˆ’2sinโกxf'(x) = -2\sin x.

Common Mistake

Forgetting the negative sign when differentiating cosโกx\cos x is a very common error. Double-check your signs!

Derivative of exe^x

This is the easiest one! The derivative of exe^x is just... exe^x. Yes, it stays the same! ๐Ÿคฏ

Example:

f(x)=ex+3x4f(x) = e^x + 3x^4

Let's find the derivative:

  • The derivative of exe^x is exe^x.
  • The derivative of 3x^4 is 12x^3 (using the power rule).

So, fโ€ฒ(x)=ex+12x3f'(x) = e^x + 12x^3.

Quick Fact

exe^x is the only function that is its own derivative. It's special!

Derivative of lnโกx\ln x

The derivative of lnโกx\ln x is 1x\frac{1}{x}. Got it? ๐Ÿ˜‰

Example:

f(x)=5lnโกx+2xf(x) = 5\ln x + 2x

Let's find fโ€ฒ(x)f'(x):

  • The derivative of 5\ln x is 5x\frac{5}{x} (since the derivative of lnโกx\ln x is 1x\frac{1}{x}).
  • The derivative of 2x is 2.

Thus, fโ€ฒ(x)=5x+2f'(x) = \frac{5}{x} + 2.

Memory Aid

Think of "ln" as "one over," then put the x in the denominator. Derivative of lnโกx\ln x is 1x\frac{1}{x}.

Final Exam Focus

Okay, let's talk strategy for the exam. Hereโ€™s what you need to focus on:

  • Memorize the Basic Derivatives: The derivatives of sinโกx\sin x, cosโกx\cos x, exe^x, and lnโกx\ln x 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 xx (e.g., sinโก(2x)\sin(2x) or ex2e^{x^2}).

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 cosโกx\cos x 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

  1. What is the derivative of f(x)=3sinโกxโˆ’2exf(x) = 3\sin x - 2e^x? (A) 3\cos x - 2e^x (B) โˆ’3cosโกxโˆ’2ex-3\cos x - 2e^x (C) 3\cos x + 2e^x (D) โˆ’3cosโกx+2ex-3\cos x + 2e^x

  2. If g(x)=4cosโกx+lnโกxg(x) = 4\cos x + \ln x, what is gโ€ฒ(x)g'(x)? (A) 4\sin x + \frac{1}{x} (B) โˆ’4sinโกx+1x-4\sin x + \frac{1}{x} (C) 4\sin x - \frac{1}{x} (D) โˆ’4sinโกxโˆ’1x-4\sin x - \frac{1}{x}

  3. Find the derivative of h(x)=5exโˆ’2sinโกx+7h(x) = 5e^x - 2\sin x + 7? (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 f(x)=2sinโกxโˆ’3cosโกx+exf(x) = 2\sin x - 3\cos x + e^x

(a) Find fโ€ฒ(x)f'(x). (2 points) (b) Find the slope of the tangent line to the graph of ff at x=0x=0. (2 points) (c) Find the equation of the tangent line to the graph of ff at x=0x=0. (3 points) (d) Find fโ€ฒโ€ฒ(x)f''(x). (2 points)

Answer Key

Multiple Choice Answers

  1. (A) 3\cos x - 2e^x
  2. (B) โˆ’4sinโกx+1x-4\sin x + \frac{1}{x}
  3. (C) 5e^x - 2\cos x

Free Response Question

(a) fโ€ฒ(x)=2cosโกx+3sinโกx+exf'(x) = 2\cos x + 3\sin x + e^x (2 points - 1 point for each correct derivative) (b) fโ€ฒ(0)=2cosโก(0)+3sinโก(0)+e0=2(1)+3(0)+1=3f'(0) = 2\cos(0) + 3\sin(0) + e^0 = 2(1) + 3(0) + 1 = 3 (2 points - 1 point for correct substitution, 1 point for correct answer) (c) f(0)=2sinโก(0)โˆ’3cosโก(0)+e0=0โˆ’3+1=โˆ’2f(0) = 2\sin(0) - 3\cos(0) + e^0 = 0 - 3 + 1 = -2. The tangent line is yโˆ’(โˆ’2)=3(xโˆ’0)y - (-2) = 3(x - 0) or y=3xโˆ’2y = 3x - 2 (3 points - 1 point for correct f(0)f(0), 1 point for correct slope, 1 point for correct equation) (d) fโ€ฒโ€ฒ(x)=โˆ’2sinโกx+3cosโกx+exf''(x) = -2\sin x + 3\cos x + e^x (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! ๐Ÿ€