# Difference rule of Differentiation

## Formula

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

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

### Introduction

The derivative of difference of any two functions is often required to calculate in differential calculus in some cases. Actually, it is impossible to find the derivative of two different functions directly. However, it can be calculated from its equivalent operation by calculating the difference of their derivatives.

$f{(x)}$ and $g{(x)}$ are two functions in terms of a variable $x$ and the derivative of difference of them can be calculated by the difference of their derivatives.

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

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

#### Leibniz’s notation

$(1) \,\,\,$ $\dfrac{d}{dx}{\, (u-v)}$ $\,=\,$ $\dfrac{du}{dx}$ $-$ $\dfrac{dv}{dx}$

#### Differentials notation

$(2) \,\,\,$ ${d}{\, (u-v)}$ $\,=\,$ $du$ $-$ $dv$

### Proof

Learn how to derive the difference rule of derivatives by first principle in differential calculus.

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