Differentiating Inverse Functions
Hannah Hill
7 min read
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Study Guide Overview
This study guide covers differentiating inverse functions, a key concept in AP Calculus AB/BC. It explains the inverse derivative formula, provides visual aids and memory aids, and offers practice problems involving direct calculations and table values. The guide also includes exam tips, common mistakes to avoid, and practice questions for both multiple-choice and free-response formats. Key topics include finding inverse functions, applying the inverse derivative formula, and writing tangent line equations.
#AP Calculus AB/BC: Differentiating Inverse Functions
Hey there, future calculus master! ๐ Let's dive into differentiating inverse functions. This is a crucial skill, and we'll make sure you're totally confident with it by the end of this guide.
This topic is frequently tested on both Multiple Choice and Free Response Questions. Make sure you understand the core concept and practice different types of problems.
# ๐ Differentiating Inverse Functions
#The Core Concept
If is the inverse of a differentiable and invertible function , then the derivative of the inverse function is given by:
rac{d}{dx}f^{-1}(x) = rac{1}{f'(f^{-1}(x))}
Or, if is the inverse of :
rac{d}{dx}g(x) = rac{1}{f'(g(x))}
Key Point: Remember, "the derivative of the inverse is the reciprocal of the derivative." This is because if , then .
Memory Aid: Think of it like a seesaw. When you differentiate the inverse, you flip the derivative of the original function. The input of the original derivative is the inverse function.
#Visualizing Inverse Functions
Graph created with Desmos
Quick Fact: Inverse functions are reflections over the line y = x. This can be helpful to visualize what the inverse function looks like.
# ๐งฎ Practice Problems
Let's solidify your understanding with a couple of e...
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