#AP Calculus AB/BC: The Power Rule - Your Ultimate Guide

Hey there, future calculus master! 👋 Let's get you prepped and confident for the AP exam with a deep dive into the Power Rule. This is a major key 🔑, so let's make sure it's locked down tight.

#The Power Rule: Your Derivative Shortcut

#What is it?

The Power Rule is your best friend for finding derivatives of power functions. Instead of dealing with those messy limits, we've got a slick shortcut. If you have a function in the form of $f(x) = x^n$, where 'n' is any constant, here's how to find its derivative:

$$f'(x) = n \cdot x^{(n-1)}$$

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• Drop it Down: Bring the exponent down to the front as a coefficient.
• Knock it Down: Reduce the original exponent by 1. ###

The derivative of any constant is always zero. This is super important when dealing with polynomials!

#Practice Problems: Let's Get Our Hands Dirty! 🏋️‍♂️

Time to put the Power Rule into action. Remember to rewrite functions if necessary before applying the rule. Let's work through these together:

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 $f'(x)$.

#Answers and Explanations: Unlocking the Solutions 👀

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

2. Problem: $f(x) = \frac{1}{x^5}$ Solution: First, rewrite $f(x) = x^{-5}$. Then, $f'(x) = -5 \cdot x^{(-5-1)} = -5x^{-6} = \frac{-5}{x^6}$

3. Problem: $f(x) = \sqrt{x}$ Solution: First, rewrite $f(x) = x^{\frac{1}{2}}$. Then, $f'(x) = \frac{1}{2} \cdot x^{\frac{1}{2}-1} = \frac{1}{2}x^{\frac{-1}{2}} = \frac{1}{2\sqrt{x}}$

4. Problem: $f(x) = x^6 + 2x^4 - 10$ Solution: $f'(x) = 6x^5 + 8x^3 + 0 = 6x^5 + 8x^3$

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Practice Question

Practice Questions

#Multiple Choice Questions

1. If $f(x) = 3x^4 - 2x^2 + 7$, then $f'(x)$ is: (A) $12x^3 - 4x + 7$ (B) $12x^3 - 4x$ (C) $7x^3 - 4x$ (D) $12x^4 - 4x^2$

2. The derivative of $g(x) = \frac{4}{x^3}$ is: (A) $\frac{12}{x^2}$ (B) $\frac{-12}{x^4}$ (C) $\frac{-4}{3x^2}$ (D) $\frac{4}{3x^2}$

#Free Response Question

Consider the function $h(x) = 2x^3 - 5\sqrt{x} + 6$.

(a) Rewrite $h(x)$ using exponents instead of radicals. (b) Find $h'(x)$. (c) Determine the slope of the tangent line to the graph of $h(x)$ at $x = 1$.

Answer Key

Multiple Choice

1. (B)
2. (B)

Free Response

(a) $h(x) = 2x^3 - 5x^{\frac{1}{2}} + 6$ (1 point) (b) $h'(x) = 6x^2 - \frac{5}{2}x^{\frac{-1}{2}}$ (2 points: 1 for each term) (c) $h'(1) = 6(1)^2 - \frac{5}{2}(1)^{\frac{-1}{2}} = 6 - \frac{5}{2} = \frac{7}{2}$ (1 point)

#Final Exam Focus 🎯

• Master the Basics: The Power Rule is fundamental and appears in almost every derivative problem.
• Rewriting Functions: Practice rewriting functions with radicals and fractions as powers. It's a crucial step for applying the Power Rule correctly.
• Combine with Other Rules: The Power Rule is often used with other rules like the product rule, quotient rule, and chain rule. Make sure you understand how they work together.

Let's move on to the next exciting topic! You're doing great! 👍