Assume, $x$ is a variable. The derivative of a variable $x$ with respect to $x$ is written in mathematical form as follows in differential calculus.

$\dfrac{d}{dx}{\, (x)}$

Use definition of the derivative to express the differentiation of a function $f{(x)}$ with respect to $x$ in limits form. It is useful to prove the differentiation of variable $x$ by first principle.

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

Take $f{(x)} \,=\, x$, then $f{(x+\Delta x)} \,=\, x+\Delta x$. Now, replace them in the above formula.

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

Take $\Delta x = h$ and write the equation in terms of $h$ in stead of $\Delta x$.

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

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

In numerator, there are three terms but two of them have opposite signs. So, they both get cancelled as per subtraction of the terms.

$\implies$ $\dfrac{d}{dx}{\, (x)}$ $\,=\,$ $\require{cancel} \displaystyle \large \lim_{h \,\to \, 0}{\normalsize \dfrac{\cancel{x}-\cancel{x}+h}{h}}$

$\implies$ $\dfrac{d}{dx}{\, (x)}$ $\,=\,$ $\displaystyle \large \lim_{h \,\to \, 0}{\normalsize \Big(\dfrac{h}{h}\Big)}$

$\implies$ $\dfrac{d}{dx}{\, (x)}$ $\,=\,$ $\require{cancel} \displaystyle \large \lim_{h \,\to \, 0}{\normalsize \Big(\dfrac{\cancel{h}}{\cancel{h}}\Big)}$

The quotient of $h$ by $h$ is equal to one mathematically.

$\implies$ $\dfrac{d}{dx}{\, (x)}$ $\,=\,$ $\displaystyle \large \lim_{h \,\to \, 0}{\normalsize \Big(1\Big)}$

There is no $h$ term in function. So, the limit of one as $h$ approaches zero is equal to one.

$\,\,\, \therefore \,\,\,\,\,\,$ $\dfrac{d}{dx}{\, (x)}$ $\,=\,$ $1$

The derivative of a variable rule is derived from first principle in this way in differential calculus.

Latest Math Topics

Dec 13, 2023

Jul 20, 2023

Jun 26, 2023

Latest Math Problems

Jan 30, 2024

Oct 15, 2023

Copyright © 2012 - 2023 Math Doubts, All Rights Reserved