# Derivative of Tangent

We shall prove the formula for the derivative of the tangent function by using definition or the first principle method.

Let us suppose that the function is of the form $y = f\left( x \right) = \tan x$.

First we take the increment or small change in the function:

Putting the value of function $y = \tan x$ in the above equation, we get

Using the formula from trigonometry, we have

Using this formula in equation (i), we get

Dividing both sides by $\Delta x$, we get

Taking the limit of both sides as $\Delta x \to 0$, we have

Example: Find the derivative of

We have the given function as

Differentiating with respect to variable $x$, we get

Using the rule, $\frac{d}{{dx}}\left( {\tan x} \right) = {\sec ^2}x$, we get