Answer
a) ${\frac{dV}{dt}}={\pi r^2 \frac{dh}{dt}}$;
b) ${\frac{dV}{dt}}={2\pi r h \frac{dr}{dt}}$;
and
c) ${\frac{dV}{dt}}={\pi r^2 \frac{dh}{dt}}+{2\pi r h \frac{dr}{dt}}$
Work Step by Step
Volume of a cylinder, $V={\pi r^2 h}$
a) Differentiate both sides, as keeping r constant, then
${\dfrac{dV}{dt}}={\pi r^2 \dfrac{dh}{dt}}$
b) When we differentiate both sides, keeping $h$ as constant:
Then ${\dfrac{dV}{dt}}={2\pi r h \dfrac{dr}{dt}}$
c) Now, when we differentiate both sides, treating both $r$ and $h$ as variables:
Thus, ${\dfrac{dV}{dt}}={\pi r^2 \dfrac{dh}{dt}}+{2\pi r h \dfrac{dr}{dt}}$
