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

Let us suppose that the function 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 formula from trigonometry, we have

Using this formula in equation (i), we get

Dividing both sides by , we get

Taking limit of both sides as , we have

Consider , as then , we get

Using the relation from limit , we have

__Example__**:** Find the derivative of

We have the given function as

Differentiation with respect to variable , we get

Using the rule, , we get