A change in a varying quantity is called a differential.

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.

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.

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 $\Delta x$.

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

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.

Latest Math Topics

Aug 31, 2024

Aug 07, 2024

Jul 24, 2024

Dec 13, 2023

Latest Math Problems

Sep 04, 2024

Jan 30, 2024

Oct 15, 2023

Copyright © 2012 - 2023 Math Doubts, All Rights Reserved