Sum rule in differentiation

Part of a series of articles about

Calculus

* Fundamental theorem

* Limits of functions
* Continuity

* Mean value theorem
* Rolle's theorem

Differential

|  Definitions

* Derivative (generalizations)

* Differential
* infinitesimal
* of a function
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Concepts

* Differentiation notation
* Second derivative
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* Related rates
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* Sum
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* General Leibniz
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Integral

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Definitions

* Antiderivative
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* Riemann integral
* Lebesgue integration
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Integration by

* Parts
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Series

* Geometric (arithmetico-geometric)
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Convergence tests

* Summand limit (term test)
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Vector

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Specialized

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Glossary of calculus

* Glossary of calculus

* v
* t
* e

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:

ddx(u+v)=dudx+dvdx

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

ddx(u1+u2+⋯+un)=du1dx+du2dx+⋯+dundx.

## 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).

h′(x)=limΔx->0h(x+Δx)−h(x)Δx=limΔx->0[f(x+Δx)+g(x+Δx)]−[f(x)+g(x)]Δx=limΔx->0f(x+Δx)−f(x)+g(x+Δx)−g(x)Δx=limΔx->0f(x+Δx)−f(x)Δx+limΔx->0g(x+Δx)−g(x)Δx=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

* Gilbert Strang: Calculus. SIAM 1991, ISBN 0-9614088-2-0, p. 71 (restricted online version (google books))
* sum rule at PlanetMath

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