Differentiation of Composite & Inverse Functions

Emily Davis

5 min read

Next Topic - Derivatives of Inverse Trigonometric Functions

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Study Guide Overview

This study guide covers the reciprocal of a derivative and derivatives of inverse functions. It explains the relationship between f'(x) and (f⁻¹)'(x), including the derivation and application of the Inverse Function Theorem. The guide provides worked examples, practice questions, and a glossary of key terms like derivative, inverse function, and chain rule. Key concepts include finding the derivative of an inverse function and relating rates of change.

#Study Notes: The Reciprocal of a Derivative and Derivatives of Inverse Functions

#Table of Contents

The Reciprocal of a Derivative
- Concept Overview
- Useful Applications
Derivatives of Inverse Functions
- Relationship between Function and Inverse Derivatives
- Deriving the Inverse Function Theorem
Worked Examples
Practice Questions
Glossary
Summary and Key Takeaways

#The Reciprocal of a Derivative

#Concept Overview

Derivatives are not fractions, but they behave similarly when finding reciprocals. The reciprocal of a derivative is crucial in various calculations, especially involving inverse functions.

$\frac{1}{\left( \frac{dy}{dx} \right)} = \frac{dx}{dy}$

This holds true as long as $\frac{dy}{dx} \neq 0$ .

Likewise, the reciprocal of $\frac{dx}{dy}$ is:

$\frac{1}{\left( \frac{dx}{dy} \right)} = \frac{dy}{dx}$

This is valid if $\frac{dx}{dy} \neq 0$ .

#Useful Applications

Key Concept

**Finding the derivative of the inverse of ...

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Previous Topic - The Chain Rule Next Topic - Derivatives of Inverse Trigonometric Functions