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

Benjamin Wright

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), eˣ, 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 🚀

Hey there! Let's dive into the derivatives of those special functions you'll see all the time: sin⁡x\sin x, cos⁡x\cos x, exe^x, and ln⁡x\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 – it's your secret weapon! 😉

FunctionDerivative
f(x)=sin⁡xf(x) = \sin xf′(x)=cos⁡xf'(x) = \cos x
g(x)=cos⁡xg(x) = \cos xg′(x)=−sin⁡xg'(x) = -\sin x
h(x)=exh(x) = e^xh′(x)=exh'(x) = e^x
k(x)=ln⁡xk(x) = \ln xk′(x)=1xk'(x) = \frac{1}{x}
Key Concept

Memorize these! They’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 sin⁡x\sin x

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

Example:

f(x)=4sin⁡x+3xf(x) = 4\sin x + 3x

To find f′(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 sin⁡x\sin x is cos⁡x\cos x).
  • The derivative of 3x is 3 (using the power rule).

So, f′(x)=4cos⁡x+3f'(x) = 4\cos x + 3.

Exam Tip

Remember to apply the constant multiple rule when a function...

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