Related rates questions have a reputation for being one of the harder topics in Extension 2 — but in three years of marking practice papers, the actual differentiation is almost never where students lose marks. It's everything before the differentiation. Here are the six mistakes I see most often, in order of how many marks they cost.
1. Not drawing (or mislabelling) the diagram
If a question mentions a ladder, a shadow, a cone filling with water, or two boats moving apart — draw it, and label every quantity that changes with a variable. Students who skip this step almost always mix up which side of the triangle is which, particularly with ladder-against-wall problems where height and base get swapped under pressure.
2. Differentiating before establishing the constraint equation
The whole method depends on writing an equation that relates the two quantities before you differentiate — usually Pythagoras', similar triangles, or a volume formula. Rushing to differentiate a quantity you haven't actually related to anything is the single most common reason a related rates question stalls halfway through.
3. Forgetting the chain rule links to time
Every rate in these questions is really a rate with respect to time, even when time doesn't
appear explicitly in your constraint equation. When you differentiate both sides, you need
d/dt, which pulls in the chain rule for every variable — this is where a lot of
dV/dt, dh/dt notation gets dropped or misapplied.
4. Substituting numbers too early
Differentiate first, substitute the specific values (radius, height, angle) second. Plugging in numbers before you differentiate turns a variable into a constant, and its derivative silently becomes zero — which quietly destroys the entire relationship you were trying to use.
5. Not checking the sign of the answer
Markers do check this. If a shadow is shrinking, its rate of change should come out negative. If water level is rising, the rate should be positive. A related rates answer with the wrong sign is an immediate signal to a marker that something upstream went wrong, even if the magnitude is correct.
6. Ignoring units (your secret setup hint)
Units in related rates questions are giant clues telling you exactly what rates you've been given or asked to find. For example, if a rate is given in cm³/s, the cm³ signals volume (V) and the /s signals time (t) — so you instantly know the question is giving you dV/dt. If a rate is in rad/s, it's an angular rate (dθ/dt). Watching the units removes all guesswork when identifying your variables.
The fix: a three-step habit
- Draw, label, and check units — let units like
cm³/stell you immediately if you're dealing withdV/dtordh/dt. - Relate before you differentiate — write the constraint equation in terms of your variables only.
- Differentiate with respect to time, then substitute known values last, and sanity-check the sign.
Run through five or six past HSC related rates questions using just these three steps, out loud, before you touch a calculator. Most students find the "maths" was never actually the issue.