Sum rule in differentiation

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:

${\displaystyle \frac{d}{dx}(u+v)=\frac{du}{dx}+\frac{dv}{dx}}$

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

${\displaystyle \frac{d}{dx}(u_1+u_2+\cdots +u_n)=\frac{d{u}_1}{dx}+\frac{d{u}_2}{dx}+\cdots +\frac{d{u}_n}{dx}.}$

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).

${\displaystyle \begin{array}{rl}{h}^{\prime }(x)& =\underset{\mathrm{\Delta }x\to 0}{\lim }\frac{h(x+\mathrm{\Delta }x)-h(x)}{\mathrm{\Delta }x}\\ & =\underset{\mathrm{\Delta }x\to 0}{\lim }\frac{[f(x+\mathrm{\Delta }x)+g(x+\mathrm{\Delta }x)]-[f(x)+g(x)]}{\mathrm{\Delta }x}\\ & =\underset{\mathrm{\Delta }x\to 0}{\lim }\frac{f(x+\mathrm{\Delta }x)-f(x)+g(x+\mathrm{\Delta }x)-g(x)}{\mathrm{\Delta }x}\\ & =\underset{\mathrm{\Delta }x\to 0}{\lim }\frac{f(x+\mathrm{\Delta }x)-f(x)}{\mathrm{\Delta }x}+\underset{\mathrm{\Delta }x\to 0}{\lim }\frac{g(x+\mathrm{\Delta }x)-g(x)}{\mathrm{\Delta }x}\\ & =f^{\prime }(x)+g^{\prime }(x).\end{array}}$

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]}

• 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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