Math Doubts

Proof of Difference rule of Differentiation

The difference rule of derivatives is actually derived in differential calculus from first principle. For example, $f{(x)}$ and $g{(x)}$ are two differentiable functions and the difference of them is written as $f{(x)}-g{(x)}$. The derivative of difference of two functions with respect to $x$ is written in the following mathematical form.

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

Define the derivative in limiting operation

Take $m{(x)} = f{(x)}-g{(x)}$ and then $m{(x+\Delta x)} = f{(x+\Delta x)}-g{(x+\Delta x)}$

According to definition of the derivative, write the derivative of the function $m{(x)}$ with respect to $x$ in limiting operation.

$\dfrac{d}{dx}{\, \Big(m{(x)}\Big)}$ $\,=\,$ $\displaystyle \large \lim_{\Delta x \,\to\, 0}{\normalsize \dfrac{m{(x+\Delta x)}-m{(x)}}{\Delta x}}$

Replace the actual functions of $m{(x)}$ and $m{(x+\Delta x)}$.

$\implies$ $\dfrac{d}{dx}{\, \Big(f{(x)}-g{(x)}\Big)}$ $\,=\,$ $\displaystyle \large \lim_{\Delta x \,\to\, 0}{\normalsize \dfrac{\Big(f{(x+\Delta x)}-g{(x+\Delta x)}\Big)-\Big(f{(x)}-g{(x)}\Big)}{\Delta x}}$

Simplify the function

Now, take $\Delta x = h$ and start simplifying this function for deriving the derivative of difference of two functions by first principle.

$=\,\,\,$ $\displaystyle \large \lim_{h \,\to\, 0}{\normalsize \dfrac{\Big(f{(x+h)}-g{(x+h)}\Big)-\Big(f{(x)}-g{(x)}\Big)}{h}}$

$=\,\,\,$ $\displaystyle \large \lim_{h \,\to\, 0}{\normalsize \dfrac{f{(x+h)}-g{(x+h)}-f{(x)}+g{(x)}}{h}}$

$=\,\,\,$ $\displaystyle \large \lim_{h \,\to\, 0}{\normalsize \dfrac{f{(x+h)}-f{(x)}-\Big(g{(x+h)}-g{(x)}\Big)}{h}}$

$=\,\,\,$ $\displaystyle \large \lim_{h \,\to\, 0}{\normalsize \Bigg[\dfrac{f{(x+h)}-f{(x)}}{h}-\dfrac{g{(x+h)}-g{(x)}}{h}\Bigg]}$

Evaluate the Limits of the functions

As per difference rule of limits, the limit of difference of two functions can be written as difference of their limits.

$=\,\,\,$ $\displaystyle \large \lim_{h \,\to\, 0}{\normalsize \dfrac{f{(x+h)}-f{(x)}}{h}}$ $-$ $\displaystyle \large \lim_{h \,\to\, 0}{\normalsize \dfrac{g{(x+h)}-g{(x)}}{h}}$

According to first principle of differentiation, each term in the right right-hand side of the equation represents the derivative of the respective function.

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

In this way, the difference rule of derivatives can be derived in differential calculus mathematically from first principle.

The derivative difference rule is also written in two forms alternatively by taking $u = f{(x)}$ and $v = g{(x)}$.

Leibniz’s notation

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

Differentials notation

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

Math Questions

The math problems with solutions to learn how to solve a problem.

Learn solutions

Math Worksheets

The math worksheets with answers for your practice with examples.

Practice now

Math Videos

The math videos tutorials with visual graphics to learn every concept.

Watch now

Subscribe us

Get the latest math updates from the Math Doubts by subscribing us.

Learn more

Math Doubts

A free math education service for students to learn every math concept easily, for teachers to teach mathematics understandably and for mathematicians to share their maths researching projects.

Copyright © 2012 - 2023 Math Doubts, All Rights Reserved