Interpret the derivative as an instantaneous rate of change and use the chain rule to relate the rates of change of connected quantities

What this dot point is asking

SCSA Unit 3 frames the derivative as a rate of change and then connects rates of related quantities. This dot point is examined in both sections, with related-rates problems appearing most often in the calculator-assumed section.

Average versus instantaneous rate

The average rate of change of $y$ over an interval is the gradient of the chord, $\dfrac{\Delta y}{\Delta x}$. The instantaneous rate is the gradient of the tangent, the limit of the chord gradient, which is the derivative. When a question asks for the rate at a specific instant, evaluate the derivative at that point.

Related rates through the chain rule

When two variables are linked by an equation and both change with time, their rates are connected by the chain rule. If $y$ depends on $x$ and $x$ depends on $t$, then

This lets you transfer a known rate into an unknown one.

Choosing the right link

The key skill is identifying the equation that connects the quantities (here $A=\pi r^{2}$) and the rate that is given (here $\dfrac{dr}{dt}$). Differentiate the relationship with respect to $t$ first, and only substitute the specific values at the very end, so that variable quantities are not frozen too early.