$(1) \,\,\,\,\,\,$ $\dfrac{d}{dx}{\, f{(x)}}$ $\,=\,$ $\displaystyle \large \lim_{\Delta x \,\to\, 0}{\normalsize \dfrac{f{(x+\Delta x)}-f{(x)}}{\Delta x}}$

$(2) \,\,\,\,\,\,$ $\dfrac{d}{dx}{\, f{(x)}}$ $\,=\,$ $\displaystyle \large \lim_{h \,\to\, 0}{\normalsize \dfrac{f{(x+h)}-f{(x)}}{h}}$

It is a formal definition of the derivative of a function in limit form, defined by limiting operation. It is written mathematically in two ways in calculus but they both are same. $\Delta x$ represents the change in variable $x$ (differential) and it is simply denoted by $h$. So, don’t get confused and you can use any one of them to find the derivative of a function in mathematics.

The principle is used as a formula to find the derivative of a function. Therefore, the method of finding derivative of a function by this rule is called in the following three ways in differential calculus.

- Derivative by first principle
- Differentiation from first principle
- ab-initio method of differentiation

If a variable is denoted by $x$, then the function in terms of $x$ is defined as $f{(x)}$ in mathematical form. The derivative of $f{(x)}$ with respect to $x$ is written in mathematics as $\dfrac{d}{dx}{\, f{(x)}}$. It is also written as $\dfrac{d{f{(x)}}}{dx}$ simply.

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

Remember, there is no rule to write this formula in terms of $x$ and $f{(x)}$ always. It can be written in terms of any variable and function but principle is same.

$\dfrac{d\, g{(t)}}{dt}$ $\,=\,$ $\displaystyle \large \lim_{\Delta t \,\to\, 0}{\normalsize \dfrac{g{(t+\Delta t)}-g{(t)}}{\Delta t}}$ $\,=\,$ $\displaystyle \large \lim_{z \,\to\, 0}{\normalsize \dfrac{g{(t+z)}-g{(t)}}{z}}$

In this case, $t$ is variable and $g{(t)}$ is a function in terms of $t$. The letter $z$ represents the change in variable $t$.

Latest Math Topics

Mar 21, 2023

Feb 25, 2023

Feb 17, 2023

Feb 10, 2023

Latest Math Problems

Mar 03, 2023

Mar 01, 2023

Feb 27, 2023

A best free mathematics education website for students, teachers and researchers.

Learn each topic of the mathematics easily with understandable proofs and visual animation graphics.

Learn how to solve the math problems in different methods with understandable steps and worksheets on every concept for your practice.

Copyright © 2012 - 2022 Math Doubts, All Rights Reserved