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.
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