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Applying the Power Rule

Abigail Young

Abigail Young

5 min read

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Study Guide Overview

This guide covers the Power Rule for finding derivatives in calculus. It explains the rule (f(x)=nx(n1)f'(x) = n * x^(n-1)), provides a mnemonic ("Drop it Down, Knock it Down"), and emphasizes the derivative of a constant is zero. Practice problems involving various function forms (polynomials, fractions, radicals) are included with solutions and explanations. The guide also highlights common mistakes and provides multiple-choice and free-response practice questions with an answer key. Finally, it offers exam tips focusing on rewriting functions and combining the Power Rule with other calculus rules.

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)=xnf(x) = x^n, where 'n' is any constant, here's how to find its derivative:

f(x)=nx(n1)f'(x) = n \cdot x^{(n-1)}

Key Concept

Key Insight: You bring the exponent down, multiply it by the existing coefficient (which is 1 in the basic case), and then reduce the exponent by one. It's like a little dance! 💃

Memory Aid

Power Rule Mnemonic: "Drop it Down, Knock it Down"

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

Quick Fact: Constants Vanish

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)=x4f(x) = x^4, find f(x)f'(x).
  2. Given f(x)=1x5f(x) = \frac{1}{x^5}, find f(x)f'(x).
  3. Given f(x)=xf(x) = \sqrt{x}, find f(x)f'(x).
  4. Given f(x)=x6+2x410f(x) = x^6 + 2x^4 - 10, find f(x)f'(x).
Exam Tip

Exam Tip: Always rewrite radicals and fractions as powers before differentiating. This avoids common errors.

Answers and Explanations: Unlocking the Solutions 👀

  1. Problem: f(x)=x4f(x) = x^4 Solution: f(x)=4x(41)=4x3f'(x) = 4 \cdot x^{(4-1)} = 4x^3

  2. Problem: f(x)=1x5f(x) = \frac{1}{x^5} Solution: First, rewrite f(x)=x5f(x) = x^{-5}. Then, f(x)=5x(51)=5x6=5x6f'(x) = -5 \cdot x^{(-5-1)} = -5x^{-6} = \frac{-5}{x^6}

  3. Problem: f(x)=xf(x) = \sqrt{x} Solution: First, rewrite f(x)=x12f(x) = x^{\frac{1}{2}}. Then, f(x)=12x121=12x12=12xf'(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)=x6+2x410f(x) = x^6 + 2x^4 - 10 Solution: f(x)=6x5+8x3+0=6x5+8x3f'(x) = 6x^5 + 8x^3 + 0 = 6x^5 + 8x^3

Common Mistake

Common Mistake: Forgetting to reduce the exponent by one, or incorrectly applying the rule to constants.

Practice Question

Practice Questions

Multiple Choice Questions

  1. If f(x)=3x42x2+7f(x) = 3x^4 - 2x^2 + 7, then f(x)f'(x) is: (A) 12x34x+712x^3 - 4x + 7 (B) 12x34x12x^3 - 4x (C) 7x34x7x^3 - 4x (D) 12x44x212x^4 - 4x^2

  2. The derivative of g(x)=4x3g(x) = \frac{4}{x^3} is: (A) 12x2\frac{12}{x^2} (B) 12x4\frac{-12}{x^4} (C) 43x2\frac{-4}{3x^2} (D) 43x2\frac{4}{3x^2}

Free Response Question

Consider the function h(x)=2x35x+6h(x) = 2x^3 - 5\sqrt{x} + 6.

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

Answer Key

Multiple Choice

  1. (B)
  2. (B)

Free Response

(a) h(x)=2x35x12+6h(x) = 2x^3 - 5x^{\frac{1}{2}} + 6 (1 point) (b) h(x)=6x252x12h'(x) = 6x^2 - \frac{5}{2}x^{\frac{-1}{2}} (2 points: 1 for each term) (c) h(1)=6(1)252(1)12=652=72h'(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.
Exam Tip

Last-Minute Tip: When you're feeling the pressure, take a deep breath, and remember "Drop it Down, Knock it Down." You've got this!

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

Question 1 of 7

What is the derivative of f(x)=x7f(x) = x^7?

7x67x^6

x6x^6

7x87x^8

6x76x^7