# What is the derivative of v=1/3pir^2h?

May 25, 2018

$\frac{\mathrm{dv}}{\mathrm{dt}} = \frac{2 \pi r h}{3} \left(\frac{\mathrm{dr}}{\mathrm{dt}}\right) + \frac{\pi {r}^{2}}{3} \left(\frac{\mathrm{dh}}{\mathrm{dt}}\right)$

#### Explanation:

if you're doing related rates, you're probably differentiating with respect to $t$ or time:
$\frac{d}{\mathrm{dt}} \left(v\right) = \frac{d}{\mathrm{dt}} \left(\frac{\pi}{3} {r}^{2} h\right)$
$\frac{\mathrm{dv}}{\mathrm{dt}} = \frac{\pi}{3} \frac{d}{\mathrm{dt}} \left({r}^{2} h\right)$
$\frac{\mathrm{dv}}{\mathrm{dt}} = \frac{\pi}{3} \left(\frac{d}{\mathrm{dt}} \left({r}^{2}\right) h + \frac{d}{\mathrm{dt}} \left(h\right) {r}^{2}\right)$
$\frac{\mathrm{dv}}{\mathrm{dt}} = \frac{\pi}{3} \left(2 r \frac{d}{\mathrm{dt}} \left(r\right) h + \frac{\mathrm{dh}}{\mathrm{dt}} {r}^{2}\right)$
$\frac{\mathrm{dv}}{\mathrm{dt}} = \frac{\pi}{3} \left(2 r \left(\frac{\mathrm{dr}}{\mathrm{dt}}\right) h + \left(\frac{\mathrm{dh}}{\mathrm{dt}}\right) {r}^{2}\right)$
$\frac{\mathrm{dv}}{\mathrm{dt}} = \frac{2 \pi r h}{3} \left(\frac{\mathrm{dr}}{\mathrm{dt}}\right) + \frac{\pi {r}^{2}}{3} \left(\frac{\mathrm{dh}}{\mathrm{dt}}\right)$