Math Doubts

Differential

A change in a varying quantity is called a differential.

Introduction

The actual meaning of a differential is difference of two quantities of a variable.

It is a basic mathematical concept in calculus and used to refer a change in a variable by a special notation. A differential can be represented either small or infinitesimal (infinitely small). Here you learn the concept of differential and its representations in calculus.

Example

$x$ is a variable. Assume, The value of $x$ at a point is $x_{1}$ and the value of $x$ at another point is $x_{2}$. The difference between them is $x_{2}-x_{1}$, which represents the change in the variable quantity $x$.

  1. If the change in $x$ is big, then it is denoted by $\small \Delta \normalsize x$.
  2. If the change in $x$ is infinitesimal (infinitely small), then it is denoted by $dx$.

For example, $x_{1} \,=\, 25$ and $x_{2} \,=\, 25.000062799\ldots$

$dx \,=\, x_{2}-x_{1}$ $\,=\,$ $25.000062799\ldots-25$

$\implies dx \,=\, 0.000062799\ldots$

The change in variable $x$ is infinitely small. It is negligible, approximately zero and cannot be measured in general. Therefore, the change in variable $x$ is denoted by $dx$ and it is called as a differential.

$y$ is a variable. Its value at a point is $30$ and its value at another point is $35$. Therefore, $y_{1} \,=\, 30$ and $y_{2} \,=\, 35$. Find the difference of them.

$y_{2}-y_{1} \,=\, 35-30$

$\implies y_{2}-y_{1} \,=\, 5$

The change in variable $y$ is not infinitesimal. Therefore, the differential is denoted by $\Delta y$

$\,\,\, \therefore \,\,\,\,\,\,\, \Delta y$ $\,=\,$ $y_{2}-y_{1} \,=\, 5$

Thus, the differentials are defined in differential calculus on the basis of change in the quantity. Remember, a differential can be denoted by anything but it represents the change in that variable.



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