# Sum rule in differentiation

Part of a series of articles about |

Calculus |
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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*:

- [math]\displaystyle{ \frac{d}{dx}(u + v)=\frac{du}{dx}+\frac{dv}{dx} }[/math]

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

- [math]\displaystyle{ \frac{d}{dx}(u_1 + u_2 + \cdots + u_n) = \frac{du_1}{dx} + \frac{du_2}{dx} + \cdots + \frac{du_n}{dx}. }[/math]

## 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*).

- [math]\displaystyle{ \begin{align} h'(x) &= \lim_{\Delta x\to 0} \frac{h(x+\Delta x)-h(x)}{\Delta x} \\ &= \lim_{\Delta x\to 0} \frac{[f(x+\Delta x)+g(x+\Delta x)]-[f(x)+g(x)]}{\Delta x} \\ &= \lim_{\Delta x\to 0} \frac{f(x+\Delta x)-f(x)+g(x+\Delta x)-g(x)}{\Delta x} \\ &= \lim_{\Delta x\to 0} \frac{f(x+\Delta x)-f(x)}{\Delta x} + \lim_{\Delta x\to 0} \frac{g(x+\Delta x)-g(x)}{\Delta x} \\ &= f'(x)+g'(x). \end{align} }[/math]

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