It is given that two functions are in form and take derivative of difference of these two functions which is equal to difference of their derivatives. This can be proving by using derivative by definition or first principle method.
Consider a 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
Dividing both sides by , we get
Taking limit of both sides as , we have
By definition of derivative we have
This shows that the derivative of difference of two given functions is equal to the difference of their derivatives.
This difference rule can be expand more than two function as
Example: Find the derivative of
We have the given function as
Differentiation with respect to variable , we get
Now using the formula derivatives, we have