Sum Rule of the Derivatives

Formula

$\dfrac{d}{dx}{\, \Big(f{(x)}+g{(x)}\Big)}$ $\,=\,$ $\dfrac{d}{dx}{\, f{(x)}}$ $+$ $\dfrac{d}{dx}{\, g{(x)}}$

The derivative of sum of functions is equal to sum of their derivatives, is called the sum rule of differentiation.

Introduction

In differential calculus, the derivative of sum of two or more functions is required to calculate in some cases. Mathematically, it is not possible to find the derivative of sum of two or more functions directly but it can be done from its equivalent mathematical operation by the sum of their derivatives.

$f{(x)}$, $g{(x)}$, $h{(x)}$ and so on are different functions in terms of a variable $x$. The derivative of sum of two or more functions can be calculated by the sum of their derivatives.

$\dfrac{d}{dx}{\, \Big(f{(x)}+g{(x)}+h{(x)}+\ldots\Big)}$ $\,=\,$ $\dfrac{d}{dx}{\, f{(x)}}$ $+$ $\dfrac{d}{dx}{\, g{(x)}}$ $+$ $\dfrac{d}{dx}{\, h{(x)}}$ $+$ $\ldots$

The sum rule of derivatives is written in two different ways popularly in differential calculus.

Leibniz’s notation

$(1) \,\,\,$ $\dfrac{d}{dx}{\, (u+v+w+\ldots)}$ $\,=\,$ $\dfrac{du}{dx}$ $+$ $\dfrac{dv}{dx}$ $+$ $\dfrac{dw}{dx}$ $+$ $\ldots$

Differentials notation

$(2) \,\,\,$ ${d}{\, (u+v+w+\ldots)}$ $\,=\,$ $du$ $+$ $dv$ $+$ $dw$ $+$ $\ldots$

Proof

Learn how to derive the property of sum rule of differentiation from first principle in differential calculus.