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➗calculus ii review

Derivative of Inverse Trigonometric Functions

Written by the Fiveable Content Team • Last updated September 2025
Written by the Fiveable Content Team • Last updated September 2025

Definition

The derivative of inverse trigonometric functions refers to the process of finding the rate of change or slope of the tangent line at a specific point on the graph of an inverse trigonometric function. This concept is particularly important in the context of integrals resulting in inverse trigonometric functions, as the derivative plays a crucial role in evaluating and simplifying these integrals.

5 Must Know Facts For Your Next Test

  1. The derivative of an inverse trigonometric function is given by the formula: $\frac{d}{dx}f^{-1}(x) = \frac{1}{f'(f^{-1}(x))}$, where $f$ is the original trigonometric function.
  2. The derivatives of the common inverse trigonometric functions are: $\frac{d}{dx}\sin^{-1}(x) = \frac{1}{\sqrt{1-x^2}}$, $\frac{d}{dx}\cos^{-1}(x) = \frac{-1}{\sqrt{1-x^2}}$, and $\frac{d}{dx}\tan^{-1}(x) = \frac{1}{1+x^2}$.
  3. The derivative of an inverse trigonometric function is often used in the evaluation of integrals that result in inverse trigonometric functions, as the derivative helps simplify the expression and make it more manageable.
  4. Understanding the derivative of inverse trigonometric functions is crucial for solving problems involving related rates, optimization, and other applications that require the use of these functions.
  5. The chain rule is an essential tool for finding the derivative of inverse trigonometric functions, as it allows you to differentiate the composite function formed by the inverse trigonometric function and another function.

Review Questions

  • Explain the general formula for the derivative of an inverse trigonometric function and how it is derived.
    • The general formula for the derivative of an inverse trigonometric function is $\frac{d}{dx}f^{-1}(x) = \frac{1}{f'(f^{-1}(x))}$, where $f$ is the original trigonometric function. This formula is derived by applying the chain rule to the composite function formed by the inverse trigonometric function and another function. The key idea is that the derivative of the inverse function is inversely proportional to the derivative of the original function evaluated at the inverse function.
  • Describe how the derivative of inverse trigonometric functions is used in the context of integrals resulting in inverse trigonometric functions.
    • When integrals result in inverse trigonometric functions, the derivative of these functions plays a crucial role in simplifying and evaluating the integrals. The derivative helps to express the inverse trigonometric function in a more manageable form, often leading to the integration of simpler functions. By applying the derivative formula for inverse trigonometric functions, the integral can be transformed into a form that is easier to integrate, allowing for the determination of the antiderivative or the evaluation of the integral.
  • Analyze the relationship between the derivatives of the common inverse trigonometric functions (sine, cosine, and tangent) and how they can be used to solve various problems in calculus.
    • The derivatives of the common inverse trigonometric functions, $\sin^{-1}(x)$, $\cos^{-1}(x)$, and $\tan^{-1}(x)$, are $\frac{1}{\sqrt{1-x^2}}$, $\frac{-1}{\sqrt{1-x^2}}$, and $\frac{1}{1+x^2}$, respectively. These derivatives are essential in solving problems involving related rates, optimization, and other applications that require the use of inverse trigonometric functions. For example, the derivative of $\tan^{-1}(x)$ can be used to find the rate of change of the angle formed by a tangent line to a curve, while the derivatives of $\sin^{-1}(x)$ and $\cos^{-1}(x)$ can be used to find the maximum or minimum values of functions involving these inverse trigonometric functions.