It is given that two functions are in form and take derivative of sum of these two functions which is equal to sum 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 sum of two given functions is equal to the sum of their derivatives.
This sum 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