Differential

A change in a varying quantity is called a differential.

Introduction

Algebraically, the varying quantity is represented by a variable and the actual meaning of a differential is the difference of two quantities of a variable.

Example

Let $x$ represents a variable, $x_1$ and $x_2$ are two values of the variable $x$. Mathematically, the difference between them is written as $x_{2}-x_{1}$, which is called the change in the variable $x$, and simply called as a differential in calculus.

Finite change

If the change in variable $x$ is finite, then the differential is denoted by $\Delta x$ in mathematics. In this case, the differential can be small or big.

$\Delta x = x_2-x_1$

For example, $x_{1} \,=\, 30$ and $x_{2} \,=\, 35$. Now, calculate the change in variable $x$.

$\implies$ $x_2-x_1 = 35-30$

$\implies$ $x_2-x_1 = 5$

The change in variable $x$ is $5$, which is finite. So, the differential is represented by $\Delta x$.

$\,\,\, \therefore \,\,\,\,\,\,$ $\Delta x = x_2-x_1 = 5$

Infinitesimal change

If the change in variable $x$ is infinitely small, then the differential is denoted by $dx$ in differential calculus. In this case, the differential is small and negligible.

$dx = x_2-x_1$

For example, $x_{1} \,=\, 21$ and $x_{2} \,=\, 21.000135$. Now, evaluate the change in variable $x$.

$\implies$ $x_2-x_1 = 21.000135-21$

$\implies$ $x_2-x_1 = 0.000135$

$\implies$ $x_2-x_1 \approx 0$

The change in variable $x$ is $0.000135$, which is infinitely small and can be negligible because its value is approximately zero. Therefore, the differential is represented by $dx$.

$\,\,\, \therefore \,\,\,\,\,\,$ $dx = x_2-x_1 = 0.000135 \approx 0$

In this way, the differentials are defined and expressed in mathematical form in differential calculus.