The derivative `(ds)/(dt)`, we mean the rate of change of distance s with respect to the time t. In a similar fashion, whenever one quantity y varies with another quantity x, satisfying some rule y= f(x) , then `(dy)/(dx)` (or f' (x)) represents the rate of change of y with respect to x and `(dy)/(dx)]_(x=x_0)` (or f′ (x0)) represents the rate of change
of y with respect to x at x = x_0 .
Further, if two variables x and y are varying with respect to another variable t, i.e., if x = f( t ) and y= g( t) , then by Chain Rule `(dy)/(dx) = (dy)/(dt)/(dx)/(dt)` , if `(dx)/(dt)` ≠ 0
Thus, the rate of change of y with respect to x can be calculated using the rate of change of y and that of x both with respect to t.
Shaalaa.com | Derivative as rate measurer 01
The surface area of a spherical balloon is increasing at the rate of 2cm2 / sec. At what rate is the volume of the balloon is increasing when the radius of the balloon is 6 cm?
If y = f (u) is a differential function of u and u = g(x) is a differential function of x, then prove that y = f [g(x)] is a differential function of x and `dy/dx=dy/(du) xx (du)/dx`
The rate of growth of bacteria is proportional to the number present. If, initially, there were
1000 bacteria and the number doubles in one hour, find the number of bacteria after 2½
[Take `sqrt2` = 1.414]