### Derivative Rules: Constant, Sum, Difference, and Constant Multiple - AP Calculus Study Guide

#### Introduction

Welcome to the magical world of derivatives, where we take functions on the ultimate thrill ride to reveal their hidden slopes! In this unit, we will uncover the secrets behind differentiating functions using the constant, sum, difference, and constant multiple rules. Brace yourselves for a fun-packed mathematical adventure! 🎢📐

#### Key Derivative Rules

#### The Constant Rule of Derivatives:

Let's start with the simplest of them all: the constant rule. Imagine you're babysitting a sleeping infant; no matter how loud you shout, they stay calm (and hopefully remain asleep). In math terms, if a function is a constant value (think of it as a sloth in a race), its derivative is zero. Mathematically, if f(x) = c, where c is a constant, then f'(x) = 0.

For example, if f(x) = 7, the derivative is f'(x) = 0. Just like a sloth, no matter how much you change the time, it remains blissfully unmoved.

#### The Sum Rule of Derivatives:

The sum rule is like organizing a fun party: the more, the merrier! If you have two functions g(x) and h(x) partying together in f(x), their derivative is just the sum of the individual derivatives. Mathematically put, if f(x) = g(x) + h(x), then f'(x) = g'(x) + h'(x).

For example, if f(x) = 2x + 5, apply the derivation individually. The derivative of 2x is 2 (thanks to our power rule friend), and the derivative of 5 (our eternally sleepy constant) is 0. Thus, f'(x) = 2 + 0 = 2. All fun, no fuss!

#### The Difference Rule of Derivatives:

Opposite to the sum rule is the difference rule, where functions decide to have a cool dance-off. If f(x) is the difference between g(x) and h(x), its derivative is the difference of their derivatives. Mathematically, if f(x) = g(x) - h(x), then f'(x) = g'(x) - h'(x).

For instance, let's say f(x) = 4x - 3. The derivative of 4x is 4, and for 3 (our sloth-like constant), the derivative is 0. Therefore, f'(x) = 4 - 0 = 4. Breakdance your way to the answer!

#### The Constant Multiple Rule of Derivatives:

So now we get to the constant multiple rule - think of it as multiplying your favorite dessert with calories. If you have a function g(x) multiplied by a constant c, the derivative is simply c times the derivative of g(x). In math terms, if f(x) = c·g(x), then f'(x) = c·g'(x).

For instance, suppose f(x) = 5(2x + 3). First, the derivative of 2x + 3 is 2 (because 3's sleepy derivative is 0). Then multiply by the 5 (let's call it a calorie bomb). So f'(x) = 5 * 2 = 10. Calories, complexity—it all multiplies!

#### Practice Problems:

Ready to flex those derivative muscles? Let's see this conceptual gym in action!

#### Derivative Example 1:

Consider the function f(x) = 3x² + 7.

Time to put on the math detective hat! Using the sum rule, identify that we can break it into g(x) = 3x² and h(x) = 7. The derivative of 3x² (using the power rule) is 6x, and for the constant 7, it's 0. Thus, f'(x) = 6x + 0 = 6x. We solved the case!

#### Derivative Example 2:

Consider the function f(x) = 42.

It's a single, bold constant! So, applying our trusty constant rule, f'(x) = 0. Nothing complicated here, just a sleeping sloth.

#### Derivative Example 3:

Consider f(x) = 4(5x + 6).

Using the constant multiple rule, find that the derivative of the inside function 5x + 6 is 5. Then, multiplying by the outer constant 4, f'(x) = 4 * 5 = 20. Pow, those multipliers!

#### Derivative Example 4:

Let’s spice it up! Consider f(x) = 7x³ - 2x² + x - 1.

Using a combo of sum, difference, and power rules:

- Derivative of 7x³ = 21x²
- Derivative of -2x² = -4x
- Derivative of x = 1
- Constant -1 drops out (derivative is zero).

So, f'(x) = 21x² - 4x + 1.

#### Combining the Power Rule with Other Derivative Rules:

Here’s the ultimate challenge! To find the derivative of a polynomial, recall:

- Apply the power rule to each term.
- Combine them using sum and difference rules.

Example function: f(x) = 2x⁵ - x⁴ + 3x² - 8:

- Derivative of 2x⁵ is 10x⁴
- Derivative of -x⁴ is -4x³
- Derivative of 3x² is 6x
- Derivative of -8 is 0 (constant).

Thus, f'(x) = 10x⁴ - 4x³ + 6x. Easy-peasy, polynomial-squeezy!

#### Final Words:

Well done, Calculus Maestro! You've aced the derivational intricacies of constant, sum, difference, and constant multiple rules like a pro. Remember, with great derivative power comes even greater AP exam scores! Now, go forth and conquer with confidence and maybe a touch of humor! 📉✨