# 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

${\displaystyle {\frac {d}{dx}}(u_{1}+u_{2}+\cdots +u_{n})={\frac {du_{1}}{dx}}+{\frac {du_{2}}{dx}}+\cdots +{\frac {du_{n}}{dx}}.}$

## 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).

{\displaystyle {\begin{aligned}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{aligned}}}

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