Differentiation of Composite & Inverse Functions

Emily Davis
6 min read
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Study Guide Overview
This study guide covers the derivatives of inverse trigonometric functions. It explains the inverse function theorem and how it's used to find the derivatives of arcsin(x), arccos(x), and arctan(x). It summarizes the derivatives of all six inverse trig functions and provides a worked example and practice problems. Key terms like the chain rule and trigonometric identities are also explained.
Derivatives of Inverse Trigonometric Functions
Table of Contents
- Introduction
- Inverse Function Theorem
- Differentiating Inverse Sine
- Differentiating Inverse Cosine
- Differentiating Inverse Tangent
- Summary of Derivatives
- Worked Example
- Practice Questions
- Glossary
- Key Takeaways
Introduction
In this section, we will explore the differentiation of inverse trigonometric functions. These functions are crucial in calculus and can be differentiated using various methods, including the inverse function theorem, the chain rule, and trigonometric identities.
Inverse Function Theorem
The inverse function theorem is essential for differentiating inverse trigonometric functions. It can be written as:
or
where .
Inverse trigonometric functions are sometimes referred to using "arc" notation, e.g., is the same as .
Differentiating Inverse Sine
To differentiate , we use the inverse function theorem:
- Let and .
- Then, .
- By the inverse function theorem:
We use the identity :
Therefore:
Thus, the derivative of is:
Ensure to use the domain restrictions for and . The derivative is defined only for .
Differentiating Inverse Cosine
To differentiate , we follow a similar process:
- Let and .
- Then, .
- By the inverse function theorem:
We use the identity :
Therefore:
Thus, the derivative of is:
Ensure to use the domain restrictions for and . The derivative is defined only for .
Differentiating Inverse Tangent
To differentiate , we use the inverse function theorem:
- Let and .
- Then, .
- By the inverse function theorem:
We use the identity :
Thus, the derivative of is:
Ensure to use the domain restrictions for and . The derivative is defined for all real numbers.
Summary of Derivatives
The table below summarizes the derivatives of all six inverse trigonometric functions:
Function | Derivative |
---|---|
-\frac{1}{ | |
\frac{1}{ | |
Worked Example
Solution:
Recall that for , we can use the chain rule.
Let and . Then .
Differentiate both functions:
Apply the chain rule:
Substitute back in:
Thus,
Practice Questions
Practice Question
1. Differentiate .
2. Differentiate .
3. Differentiate .
Glossary
- Inverse Function Theorem: A method to find the derivative of an inverse function.
- Chain Rule: A rule for differentiating compositions of functions.
- Trigonometric Identities: Equations involving trigonometric functions that are true for all values of the variables.
Key Takeaways
- The derivatives of inverse trigonometric functions are derived using the inverse function theorem, chain rule, and trigonometric identities.
- Domain restrictions are crucial for ensuring the derivatives are correctly applied.
- Practice differentiating various compositions of inverse trigonometric functions to master the concepts.
Conclusion
Understanding the derivatives of inverse trigonometric functions is essential in calculus. By mastering the inverse function theorem, chain rule, and trigonometric identities, you can confidently differentiate these functions in various contexts. Practice regularly to reinforce your knowledge and improve your problem-solving skills.

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Question 1 of 11
What is the derivative of ? 🤔