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