It is given that the derivative of a function that is the sum of two other functions, is equal to the sum of their derivatives. This can be proved by using the 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 the limit of both sides as , we have
By the definition of derivative we have
This shows that the derivative of the sum of two given functions is equal to the sum of their derivatives.
This sum rule can be expanded more than two function as
Example: Find the derivative of
We have the given function as
Differentiating with respect to variable , we get
Now using the formula of derivatives, we have