# Derivative of Cosine Inverse

In this tutorial we shall discuss the derivative of inverse trigonometric functions and first we shall prove the cosine inverse trigonometric function.

Let the function of the form be

By the definition of inverse trigonometric function, $y = {\cos ^{ - 1}}x$ can be written as

Differentiating both sides with respect to the variable $x$, we have

Since $y$ is restricted in the interval $\left] {0,\pi } \right[$ for $- 1 < x < 1$, $\sin y$ can have only positive values, and from the fundamental trigonometric rules $\sin y = \sqrt {1 - {{\cos }^2}y}$. Putting this value in the above relation (i) and simplifying, we have

Example: Find the derivative of

We have the given function as

Differentiating with respect to variable $x$, we get

Using the cosine inverse rule, $\frac{d}{{dx}}\left( {{{\cos }^{ - 1}}x} \right) = - \frac{1}{{\sqrt {1 - {x^2}} }}$, we get