Question

A stone dropped into a pond of still water sends out concentric circular waves from the point of disturbance of water at the rate of 4 cm/sec. Find the rate of change of disturbed area at the instant when the radius of the wave ring is 15 cm.

Hint: The derivative of Y with respect to X, written $\dfrac{{dy}}{{dx}}$, is just a description of how fast Y changes when X changes. It so happens that if $Y = {X^N}$, then $\dfrac{{dy}}{{dx}} = N{X^{N - 1}}$. So, for example, if $Y = 5{X^3}$, then $\dfrac{{dy}}{{dx}} = 15{X^2}$. The area of a circle is $\pi {r^2}$, and the circumference is $2\pi r$, which is the derivative.
Chain rule:
To differentiate y = f(g(x)), let u = g(x). Then y = f(u) and
$\dfrac{{dy}}{{dx}}{\text{ }} = {\text{ }}\dfrac{{dy}}{{du}}{\text{ }} \times {\text{ }}\dfrac{{du}}{{dx}}$

If the radius is increasing at a constant rate of, $\dfrac{{dr}}{{dt}} = 4cm/\sec$
Let area be $A = \pi {r^2}$…………………………(1)
Differentiating the equation (1) with respect to, $\dfrac{{dA}}{{dt}} = 2\pi r \times \dfrac{{dr}}{{dt}}$
So, Rate of change of disturbed area = $\dfrac{{dA}}{{dt}} = 2\pi r \times \dfrac{{dr}}{{dt}}$
$\Rightarrow \dfrac{{dA}}{{dt}} = 2\pi \left( {15} \right) \times \left( 4 \right) \\ \Rightarrow \dfrac{{dA}}{{dt}} = 376.99c{m^2}/\sec \\$
The rate of change of disturbed area is $376.99c{m^2}/\sec$at the instant when the radius of the wave ring is 15 cm.
Note: $\dfrac{{dy}}{{dx}}$ is positive if y increases as x increases and is negative if y decreases as x increases. The same “derivative thing” holds up for the circumference vs. the area of a circle. The change in area, $dA$, is $dA = (2\pi r)dR$. So,$\dfrac{{dA}}{{dR}} = 2\pi r$. That is, the derivative of the area is just the circumference.