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 .
First we take the increment or small change in the function:
Putting the value of function 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 , we get
Taking the limit of both sides as , we have
Example: Find the derivative of
We have the given function as
Differentiating with respect to variable , we get
Using the rule, , we get