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:

[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.

References

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