Definition of the Derivative

Formula

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

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

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

Introduction

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 $t$.