Quick questions on Implicit differentiation and related rates in QCE Specialist Mathematics Unit 4

If two quantities depend on time $t$, their rates link through the chain rule. For $V$ a function of $r$,

The crucial order is to differentiate first and substitute the specific values only afterward. Substituting a numerical value (such as a fixed radius) before differentiating treats a varying quantity as constant and loses its rate term, a frequent and heavily penalised error.

Given $\frac{dV}{dt} = 100$ and $r = 5$:

The rate is positive, consistent with the balloon expanding, and the units cm/s are correct for a rate of change of length. At a larger radius the same inflow would give a smaller $\frac{dr}{dt}$, since $4\pi r^2$ grows, which makes physical sense.