Sum rule in differentiation

From Wikipedia, the free encyclopedia

In calculus, the sum rule in differentiation is a method of finding the derivative of a function that is the sum of two other functions for which derivatives exist. This is a part of the linearity of differentiation. The sum rule in integration follows from it. The rule itself is a direct consequence of differentiation from first principles.

The sum rule states that for two functions u and v:

This rule also applies to subtraction and to additions and subtractions of more than two functions

Proof

Let h(x) = f(x) + g(x), and suppose that f and g are each differentiable at x. Applying the definition of the derivative and properties of limits gives the following proof that h is differentiable at x and that its derivative is given by h(x) = f(x) + g(x).

A similar argument shows the analogous result for differences of functions. Likewise, one can either use induction or adapt this argument to prove the analogous result for a finite sum of functions. However, the sum rule does not in general extend to infinite sums of functions unless one assumes something like uniform convergence of the sum.[citation needed]

References

Retrieved from "https://en.wikipedia.org/w/index.php?title=Sum_rule_in_differentiation&oldid=803934630"
This content was retrieved from Wikipedia : http://en.wikipedia.org/wiki/Sum_rule_in_differentiation
This page is based on the copyrighted Wikipedia article "Sum rule in differentiation"; it is used under the Creative Commons Attribution-ShareAlike 3.0 Unported License (CC-BY-SA). You may redistribute it, verbatim or modified, providing that you comply with the terms of the CC-BY-SA