Math Doubts

Derivative Rule of Inverse Cosine function


$\dfrac{d}{dx}{\, \cos^{-1}{x}} \,=\, -\dfrac{1}{\sqrt{1-x^2}}$


When $x$ represents a variable, the inverse cosine function is written as $\cos^{-1}{(x)}$ or $\arccos{(x)}$ in inverse trigonometry. In differential calculus, the derivative of the cos inverse function with respect to $x$ is written in following two mathematical forms.

$(1) \,\,\,$ $\dfrac{d}{dx}{\, \Big(\cos^{-1}{(x)}\Big)}$

$(2) \,\,\,$ $\dfrac{d}{dx}{\, \Big(\arccos{(x)}\Big)}$

The derivative of the inverse cos function with respect to $x$ is equal to the negative reciprocal of the square root of the subtraction of square of $x$ from one.

$\implies$ $\dfrac{d}{dx}{\,\Big(\cos^{-1}{(x)}\Big)}$ $\,=\,$ $-\dfrac{1}{\sqrt{1-x^2}}$

Alternative forms

The differentiation of the cos inverse function can be written in any variable. Here are few examples to learn how to write the formula for the derivative of cosine inverse function in differential calculus.

$(1) \,\,\,$ $\dfrac{d}{dz}{\,\Big(\cos^{-1}{(z)}\Big)}$ $\,=\,$ $-\dfrac{1}{\sqrt{1-z^2}}$

$(2) \,\,\,$ $\dfrac{d}{du}{\,\Big(\cos^{-1}{(u)}\Big)}$ $\,=\,$ $-\dfrac{1}{\sqrt{1-u^2}}$

$(3) \,\,\,$ $\dfrac{d}{dy}{\,\Big(\cos^{-1}{(y)}\Big)}$ $\,=\,$ $-\dfrac{1}{\sqrt{1-y^2}}$


Learn how to prove the differentiation of the inverse cosine function formula by first principle.

Math Questions

The math problems with solutions to learn how to solve a problem.

Learn solutions

Math Worksheets

The math worksheets with answers for your practice with examples.

Practice now

Math Videos

The math videos tutorials with visual graphics to learn every concept.

Watch now

Subscribe us

Get the latest math updates from the Math Doubts by subscribing us.

Learn more

Math Doubts

A free math education service for students to learn every math concept easily, for teachers to teach mathematics understandably and for mathematicians to share their maths researching projects.

Copyright © 2012 - 2023 Math Doubts, All Rights Reserved