### Derivatives of ( \cos x ), ( \sin x ), ( e^x ), and ( \ln x ): AP Calculus AB/BC Study Guide 2024

#### Introduction

Welcome to the wild ride of differentiation! Today, we're diving into the thrilling world of derivatives of some very special functions: ( \sin x ), ( \cos x ), ( e^x ), and ( \ln x ). Get ready to unlock the secrets of these functions with some humor, a bit of charm, and lots of math magic! 🧙♂️✨

#### The Magic of Special Derivatives

Let's start with a table to summarize our magical differentiations. Think of this as your spellbook for math sorcery!

| Function | Derivative | |-----------------------|---------------------| | ( f(x) = \sin x ) | ( f'(x) = \cos x )| | ( g(x) = \cos x ) | ( g'(x) = -\sin x )| | ( h(x) = e^x ) | ( h'(x) = e^x ) | | ( k(x) = \ln x ) | ( k'(x) = \frac{1}{x} )|

#### Derivative of ( \sin x )

The derivative of ( \sin x ) is ( \cos x ). This is as straightforward as realizing your dog will eat your homework given the chance.

**Example:**
Let's look at the function:
[ f(x) = 4\sin x + 3x ]

To find the derivative, differentiate each part separately. The derivative of ( 4\sin x ) is ( 4\cos x ), because multiplying by a constant is just that easy. The derivative of ( 3x ) is simply 3. Combine these, and you get: [ f'(x) = 4\cos x + 3 ]

Easy peasy, right? 😎

#### Derivative of ( \cos x )

The derivative of ( \cos x ) is ( -\sin x ). It’s like a mathematical transformation—flip that sine and make it negative!

**Example:**
Consider the function:
[ f(x) = 2\cos x + 3 ]

The derivative of ( 2\cos x ) is ( -2\sin x ) (because the derivative of ( \cos x ) is ( -\sin x ) and we multiply by 2). The derivative of the constant 3 is 0 (constants are the slackers of derivatives—they never change). So: [ f'(x) = -2\sin x ]

#### Derivative of ( e^x )

Now, prepare to be amazed. The derivative of ( e^x ) is... ( e^x )! It's the function that’s so nice it differentiates to itself. Even (e)Len DeGeneres would be impressed by such consistency!

**Example:**
Here's a fun one:
[ f(x) = e^x + 3x^4 ]

The derivative of ( e^x ) is ( e^x ). For the second part, use the power rule: the derivative of ( 3x^4 ) is ( 12x^3 ). Putting it all together: [ f'(x) = e^x + 12x^3 ]

If only life were as predictable as ( e^x ).

#### Derivative of ( \ln x )

The derivative of ( \ln x ) is ( \frac{1}{x} ). Think of it as nature’s way of saying that logarithms have important reciprocal relationships.

**Example:**
Consider:
[ f(x) = 5\ln x + 2x ]

The derivative of ( 5\ln x ) is ( \frac{5}{x} ). The derivative of ( 2x ) is 2. So combining these, we get: [ f'(x) = \frac{5}{x} + 2 ]

Voila! You’ve got a differentiated function that even Newton would applaud. 👏

#### Key Concepts to Know

To solidify your math wizardry, make sure you familiarize yourself with these key terms:

**1/x**: This is a rational function that shows an inverse relationship—kind of like trying to push a rope.**2e^(2x)**: This is an exponential function showing rapid growth or decay, like watching a viral video blow up.**Derivatives**: These measure how a function changes as its input changes. Think of them as the speedometers of the math world.**E^x**: The heavyweight champion of continuous growth, ( e^x ), is defined by raising Euler's number (approximately 2.71828) to the power of x.**Exponential Function**: Functions where the variable is in the exponent, causing impressive growth or decay. It’s like the secret sauce for modeling nature and finance.**F'(x)**: The notation for the derivative of function f(x), indicating how the function is changing at any point.**Natural Logarithm Function (ln x)**: The inverse of the exponential function. It's the mathematical version of hitting the "undo" button.**Sin x**: Refers to the sine trigonometric function, connecting angles to side lengths in right triangles—your go-to for periodic phenomena.

#### Conclusion

Congratulations! You've navigated the winding paths of special derivatives and come out the other end a math sorcerer. Remember, practice is key, and these rules will soon become as familiar as your favorite meme. Good luck with your AP Calculus, and may the derivatives be ever in your favor! 📈✨