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:

{\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

{\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 $h' (x) = f' (x) + g' (x)$ .

{\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.

References

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

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.