What Is a Related Rates of Change Problem?

A related rates of change problem asks you to find the rate at which one quantity changes by using its mathematical relationship with another changing quantity. In IB Mathematics: Applications and Interpretation, this is HL-only content under AHL 5.9 Differentiation rules.

The Mechanism

Both quantities depend on the same independent variable, usually time $t$. An equation connects the quantities, and differentiation with respect to time produces an equation connecting their rates.

Use this method:

1. Define the changing variables and their units.
2. Write an equation relating the variables.
3. Differentiate the entire equation with respect to $t$, usually using the chain rule.
4. Substitute the values that apply at the stated instant.
5. Solve for the required rate and include its units.

The rates may be positive or negative. A positive rate means the quantity is increasing, while a negative rate means it is decreasing. All rates must refer to the same instant, so do not combine a radius from one moment with a volume rate from another. A diagram can help identify the relationship, but it does not replace the equation.

For example, suppose air enters a spherical balloon at

$$\frac{dV}{dt}=12\,\mathrm{m}^3\mathrm{s}^{-1}.$$

Find the rate at which its radius increases when $r=2\,\mathrm{m}$. The sphere's volume is

$$V=\frac{4}{3}\pi r^3.$$

Differentiate with respect to time:

$$\frac{dV}{dt}=4\pi r^2\frac{dr}{dt}.$$

Now substitute $\frac{dV}{dt}=12$ and $r=2$:

$$12=4\pi(2)^2\frac{dr}{dt}.$$

Therefore,

$$\frac{dr}{dt}=\frac{3}{4\pi}\approx 0.239\,\mathrm{m\,s}^{-1}.$$

The radius is increasing at approximately $0.239\,\mathrm{m\,s}^{-1}$. The units provide a useful check: $\frac{dV}{dt}$ has cubic units per time, whereas has length units per time. If the final units are inconsistent, revisit the differentiated equation.

A common misconception is to substitute $r=2$ before differentiating. This incorrectly treats the changing radius as a constant. Differentiate first, then substitute values for the particular instant.

Exam Technique

For an IB response, show the original relationship, differentiated equation, substitution, and contextual answer with units. Although every Math AI paper permits a GDC, the correct algebraic set-up is essential for method credit.