Inverse Trigonometric Derivative Calculator

Published on: July 18, 2026
Final Answer: Free • Full Steps: Plus

This Inverse Trigonometric Derivative Calculator differentiates arcsin(x), arccos(x), and arctan(x). Each inverse trig function has its own derivative rule, and any constant multiple simply stays in front.

Step-by-step method

  1. Identify the inverse trigonometric function and any constant multiple.
  2. Write the derivative rule for that inverse function.
  3. Apply the rule, keeping the constant multiple in front.

Formula:

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

Example 1:

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

Step 1 - Identify the inverse trigonometric function and any constant multiple.

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

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

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 - 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:

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

Example 2:

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

Step 1 - Identify the inverse trigonometric function and any constant multiple.

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

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

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 - 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:

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