Connect changing quantities

Implicit differentiation treats y as a function of x even when the relation is not solved for y. Differentiate both sides and attach dy/dx whenever a term depends on y.

For related rates, first write a relation among the quantities, then differentiate with respect to time. Substitute the values for the instant of interest after differentiating, so changing quantities are not treated as constants.

\displaystyle x^2+y^2=r^2\Rightarrow\frac{dy}{dx}=-\frac xy,\quad\frac{dV}{dt}=4\pi r^2\frac{dr}{dt}

WORKED EXAMPLE

A sphere’s radius increases at 2 cm/s. Find its volume rate when r = 3 cm.

V = ( $\frac{4}{3}$ )πr³, so dV/dt = 4πr² dr/dt.
Substitute r = 3 and dr/dt = 2.
The volume rate is 72π cm³/s.

Assumed knowledge

Use the chain, product and power rules for differentiation. Rearrange relations between quantities, interpret derivative units, and recall standard area and volume formulas.

Study Specialist Mathematics alongside, or after, Mathematical Methods. Later units build on earlier units.

QCAA prior-learning guidance ↗

Connect changing quantities

A sphere’s radius increases at 2 cm/s. Find its volume rate when r = 3 cm.

Put the idea to work.