Physical meaning of \(
\frac{{dy}}
{{dx}}
\):
The concept of differentiation is made use of in physics in determining the instantaneous rate of change of a physical quantity w.r.t. some other quantity, which varies in a continuous manner.
1. The ratio of small increments in the function y and the variable x is called the average rate of change of y w.r.t.x.
If a body covers a small distance \(\Delta s\) in small time \(\Delta t\), then average velocity of the body \(
\upsilon _{av} = \frac{{\Delta S}}
{{\Delta t}}
\)
Again, if the velocity of a body changes by a small amount \(\Delta v\) in small time \(\Delta t\), then average acceleration of the body, \(
\upsilon _{av} = \frac{{\Delta \upsilon }}
{{\Delta t}}
\)
2. The limiting value of \(
\frac{{\Delta y}}
{{\Delta x}}
\), when \(\Delta x\to0\) i.e., \(
\mathop {\lim }\limits_{\Delta x \to 0} \frac{{\Delta y}}
{{\Delta x}} = \frac{{dy}}
{{dx}}
\) is called the instantaneous rate of change of y w.r.t.x.
Thus, the differentiation of a function w.r.t. a variable implies the instantaneous rate of change of the function w.r.t. that variable.
Likewise, instantaneous velocity of the body, \(
\upsilon = \mathop {\lim }\limits_{\Delta x \to 0} \frac{{\Delta S}}
{{\Delta t}} = \frac{{dS}}
{{dt}}
\) and instantaneous acceleration of the body, \(
a = \mathop {\lim }\limits_{\Delta x \to 0} \frac{{\Delta \upsilon }}
{{\Delta t}} = \frac{{d\upsilon }}
{{dt}}
\)