# Derivative of Sine Inverse

In this tutorial we shall be concerned with the derivative of inverse trigonometric functions and first we shall prove sine inverse trigonometric function.

Let the function of the form

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

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

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

Example: Find the derivative of

We have the given function as

Differentiation with respect to variable $x$, we get

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