#Derivatives of Inverse Trigonometric Functions

#Table of Contents

1. Introduction
2. Inverse Function Theorem
3. Differentiating Inverse Sine
4. Differentiating Inverse Cosine
5. Differentiating Inverse Tangent
6. Summary of Derivatives
7. Worked Example
8. Practice Questions
9. Glossary
10. 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:

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

or

$${g}^{\prime}(x) = \frac{1}{{f}^{\prime}(g(x))}$$

where $g(x) = {f}^{-1}(x)$.

#Differentiating Inverse Sine

To differentiate $\sin^{-1} x$, we use the inverse function theorem:

1. Let $g(x) = \sin^{-1} x$ and $ f(x) = \sin ...