Q.A boy is blowing air into a spherical balloon and thus the radius r of the balloon is changing, then find the rate of change of surface area of the balloon with respect to the radius r. Also find the rate of change of surface area when r = 2 cm.

Related Rates: When Two Things Change Together

Imagine you're blowing up a balloon. Your lungs push air in at a certain rate — say, 3 cubic centimetres per second. As the balloon grows, its radius increases. The question is: how fast is the radius increasing at the exact moment when the radius is 5 cm?

That's a related rates problem. Two quantities — the volume $V$ and the radius $r$ — are linked by a geometric formula ($V = \frac{4}{3}\pi r^3$). You know how fast one is changing ($dV/dt = 3$), and you want to find how fast the other is changing ($dr/dt$) at a specific instant.

The core idea is simple: if two quantities are connected by an equation, their rates of change are also connected — by the derivative of that equation.

The Precise Statement

Let $x$ and $y$ be two quantities that both depend on time $t$, and suppose they satisfy some equation $F(x, y) = 0$ (or $y = f(x)$, etc.). Then:

1. Differentiate both sides of the equation with respect to time $t$.
2. Use the chain rule wherever you see a variable that depends on $t$.
3. The result is an equation linking $dx/dt$ and $dy/dt$.

That's it. The "related rates" are $dx/dt$ and $dy/dt$, and the chain rule is the tool that connects them.

The Chain Rule in Action

In the balloon example, the equation is $V = \frac{4}{3}\pi r^3$. Differentiate both sides with respect to $t$:

The right side is a function of $r$, and $r$ itself depends on $t$. By the chain rule:

Now plug in what you know: $dV/dt = 3$ and $r = 5$:

So:

That's the answer. The radius is growing at about 0.0095 cm/s when the balloon's radius is 5 cm.

The General Recipe

For any related rates problem, follow these steps:

1. Identify all changing quantities and assign them variables. Note which rates you know and which you need.
2. Write an equation that relates the quantities at any time $t$ (not just at the instant of interest).
3. Differentiate implicitly with respect to $t$. Use the chain rule for every variable that depends on $t$.
4. Substitute the known values (including the specific instant's measurements) into the rate equation.
5. Solve for the unknown rate.

A Second Example: The Ladder Problem …