Inverse Trigonometric Derivative Notes

Published on: September 29, 2026

This Inverse Trigonometric Derivative Calculator differentiates arcsine, arccosine, arctangent and their relatives, and shows the working line by line. Each of these derivatives is an algebraic fraction rather than another trigonometric function, and the arc-co versions differ from their partners only by a minus sign. A constant multiple rides along unchanged. Following the same pattern every time keeps those paired signs from being swapped.

Step-by-step method

  1. Set up the problem.
  2. Write the derivative rule for that inverse function.
  3. Apply the rule, keeping the constant multiple in front.

Formula:

\(\begin{gathered} \frac{d}{dx}\left(\arcsin\left(x\right)\right) = \frac{1}{\sqrt{1 - x^{2}}} \\ \frac{d}{dx}\left(\arccos\left(x\right)\right) = -\frac{1}{\sqrt{1 - x^{2}}} \\ \frac{d}{dx}\left(\arctan\left(x\right)\right) = \frac{1}{1 + x^{2}} \end{gathered}\)
See Example 1 Hide Example 1

Problem

\(f\left(x\right) = \arcsin{\left(x \right)}\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the derivative rule for that inverse function.
  3. Apply the rule, keeping the constant multiple in front.

Step 1

Step 1 - Set up the problem.

In this problem: The function is \(\arcsin{\left(x \right)}\).

\(f\left(x\right) = \arcsin{\left(x \right)}\)

Step 2

Step 2 - Write the derivative rule for that inverse function.

In this problem: The derivative of \(\arcsin{\left(x \right)}\) is \(\frac{1}{\sqrt{1 - x^{2}}}\).

\(\frac{d}{dx}\left(\arcsin{\left(x \right)}\right) = \frac{1}{\sqrt{1 - x^{2}}}\)

Step 3

Step 3 - Apply the rule, keeping the constant multiple in front.

In this problem: Replace the function with its derivative and simplify.

\(f'\left(x\right) = \frac{1}{\sqrt{1 - x^{2}}}\)

Final Answer

\(f'\left(x\right) = \frac{1}{\sqrt{1 - x^{2}}}\)
See Example 2 Hide Example 2

Problem

\(f\left(x\right) = \arctan{\left(x \right)}\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the derivative rule for that inverse function.
  3. Apply the rule, keeping the constant multiple in front.

Step 1

Step 1 - Set up the problem.

In this problem: The function is \(\arctan{\left(x \right)}\).

\(f\left(x\right) = \arctan{\left(x \right)}\)

Step 2

Step 2 - Write the derivative rule for that inverse function.

In this problem: The derivative of \(\arctan{\left(x \right)}\) is \(\frac{1}{x^{2} + 1}\).

\(\frac{d}{dx}\left(\arctan{\left(x \right)}\right) = \frac{1}{x^{2} + 1}\)

Step 3

Step 3 - Apply the rule, keeping the constant multiple in front.

In this problem: Replace the function with its derivative and simplify.

\(f'\left(x\right) = \frac{1}{x^{2} + 1}\)

Final Answer

\(f'\left(x\right) = \frac{1}{x^{2} + 1}\)
See Example 3 Hide Example 3

Problem

\(f\left(x\right) = \arccos{\left(x \right)}\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the derivative rule for that inverse function.
  3. Apply the rule, keeping the constant multiple in front.

Step 1

Step 1 - Set up the problem.

In this problem: The function is \(\arccos{\left(x \right)}\).

\(f\left(x\right) = \arccos{\left(x \right)}\)

Step 2

Step 2 - Write the derivative rule for that inverse function.

In this problem: The derivative of \(\arccos{\left(x \right)}\) is \(- \frac{1}{\sqrt{1 - x^{2}}}\).

\(\frac{d}{dx}\left(\arccos{\left(x \right)}\right) = - \frac{1}{\sqrt{1 - x^{2}}}\)

Step 3

Step 3 - Apply the rule, keeping the constant multiple in front.

In this problem: Replace the function with its derivative and simplify.

\(f'\left(x\right) = - \frac{1}{\sqrt{1 - x^{2}}}\)

Final Answer

\(f'\left(x\right) = - \frac{1}{\sqrt{1 - x^{2}}}\)
See Example 4 Hide Example 4

Problem

\(f\left(x\right) = 2 \arctan{\left(x \right)}\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the derivative rule for that inverse function.
  3. Apply the rule, keeping the constant multiple in front.

Step 1

Step 1 - Set up the problem.

In this problem: The function is \(\arctan{\left(x \right)}\). The constant multiple \(2\) stays in front.

\(f\left(x\right) = 2 \arctan{\left(x \right)}\)

Step 2

Step 2 - Write the derivative rule for that inverse function.

In this problem: The derivative of \(\arctan{\left(x \right)}\) is \(\frac{1}{x^{2} + 1}\).

\(\frac{d}{dx}\left(\arctan{\left(x \right)}\right) = \frac{1}{x^{2} + 1}\)

Step 3

Step 3 - Apply the rule, keeping the constant multiple in front.

In this problem: Replace the function with its derivative and simplify.

\(f'\left(x\right) = 2 \cdot \left(\frac{1}{x^{2} + 1}\right) = \frac{2}{x^{2} + 1}\)

Final Answer

\(f'\left(x\right) = \frac{2}{x^{2} + 1}\)