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: $\sin x$, $\cos x$, $e^x$, and $\ln x$. 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! 😉

Derivative of $\sin x$

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

Example:

$$f(x) = 4\sin x + 3x$$

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

• The derivative of 4\sin x is 4\cos x (since the derivative of $\sin x$ is $\cos x$).
• The derivative of 3x is 3 (using the power rule).

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

Derivative of $\cos x$

The derivative of $\cos x$ is always $-\sin x$. Don't forget that negative sign! It's a common place to slip up. 😬

Example:

$$f(x) = 2\cos x + 3$$

Let's break it down:

• The derivative of 2\cos x is $-2\sin x$ (since the derivative of $\cos x$ is $-\sin x$).
• The derivative of the constant 3 is 0.

Therefore, $f'(x) = -2\sin x$.

Derivative of $e^x$

This is the easiest one! The derivative of $e^x$ is just... $e^x$. Yes, it stays the same! 🤯

Example:

$$f(x) = e^x + 3x^4$$

Let's find the derivative:

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

So, $f'(x) = e^x + 12x^3$.

Derivative of $\ln x$

The derivative of $\ln x$ is $\frac{1}{x}$. Got it? 😉

Example:

$$f(x) = 5\ln x + 2x$$

Let's find $f'(x)$:

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

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

Final Exam Focus

Okay, let's talk strategy for the exam. Here’s what you need to focus on:

• Memorize the Basic Derivatives: The derivatives of $\sin x$, $\cos x$, $e^x$, and $\ln x$ 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 $x$ (e.g., $\sin(2x)$ or $e^{x^2}$).

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 $\cos x$ 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 $f(x) = 3\sin x - 2e^x$? (A) 3\cos x - 2e^x (B) $-3\cos x - 2e^x$ (C) 3\cos x + 2e^x (D) $-3\cos x + 2e^x$

2. If $g(x) = 4\cos x + \ln x$, what is $g'(x)$? (A) 4\sin x + \frac{1}{x} (B) $-4\sin x + \frac{1}{x}$ (C) 4\sin x - \frac{1}{x} (D) $-4\sin x - \frac{1}{x}$

3. Find the derivative of $h(x) = 5e^x - 2\sin x + 7$? (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 $f(x) = 2\sin x - 3\cos x + e^x$

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

Answer Key

Multiple Choice Answers

1. (A) 3\cos x - 2e^x
2. (B) $-4\sin x + \frac{1}{x}$
3. (C) 5e^x - 2\cos x

Free Response Question

(a) $f'(x) = 2\cos x + 3\sin x + e^x$ (2 points - 1 point for each correct derivative) (b) $f'(0) = 2\cos(0) + 3\sin(0) + e^0 = 2(1) + 3(0) + 1 = 3$ (2 points - 1 point for correct substitution, 1 point for correct answer) (c) $f(0) = 2\sin(0) - 3\cos(0) + e^0 = 0 - 3 + 1 = -2$. The tangent line is $y - (-2) = 3(x - 0)$ or $y = 3x - 2$ (3 points - 1 point for correct $f(0)$, 1 point for correct slope, 1 point for correct equation) (d) $f''(x) = -2\sin x + 3\cos x + e^x$ (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! 🍀