2.5 Applying the Power Rule

Welcome back to AP Calculus with Fiveable! We are now diving into one of the most valuable fundamental concepts in calculus: the Power Rule. This is the first of many derivative rules that you’re going to learn about!

⚡ The Power Rule

The Power Rule states that if $f(x) = x^n$, where $n$ is constant, then the derivative $f'(x)$ is given by:

So, the Power Rule provides a shortcut for finding the derivative without using the limit definition of a derivative. Wasn’t doing that annoying?!

🏋️‍♂️ Practice Problems****

Let’s work on a few questions to get the Power Rule down!

1. Given $f(x) = x^4$, find $f'(x)$.
2. Given $f(x) = \frac{1}{x^5}$, find $f'(x)$.
3. Given $f(x) = \sqrt{x}$, find $f'(x)$.
4. Given $f(x) = x^6 +2x^4-10$, find $f'(x)$.

💡 Before we reveal the answers, remember:

1. The Power Rule with fractions can be tricky! Sometimes rewriting the equation can help.
2. The derivative of any constant is zero.

👀 Answers to Practice Problems

Note how for many of these problems, the equations were rewritten before solving for the derivative using the power rule.

1. $f'(x) = 4 \cdot x^{(4-1)}$ $= 4x^3$

2. $f(x) = x^{-5}$

   $f'(x) = -5 \cdot x^{(-5-1)}$ $= -5x^{-6}$ $= \frac{-5}{x^{6}}$

3. $f(x) = x^{1/2}$

   $f'(x) = \frac{1}{2} \cdot x^\frac{-1}{2}$ $= \frac {1}{2\sqrt{x}}$

4. $f'(x)=6x^5 + 8x^3$

Yep! That’s it. This lesson was super short. Want to jump into the rest of the derivative rules you have to know? ⏭️