# Derivative of Tangent Inverse

In this tutorial we shall explore the derivative of inverse trigonometric functions and we shall prove the derivative of tangent inverse.

Let the function of the form be

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

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

Using the fundamental trigonometric rules, we can write this as $1 + {\tan ^2}y = {\sec ^2}y$. 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, $\frac{d}{{dx}}\left( {{{\tan }^{ - 1}}x} \right) = \frac{1}{{1 + {x^2}}}$, we get