Find the rate of change of the volume of a sphere with respect to its diameter.

Question:

Find the rate of change of the volume of a sphere with respect to its diameter.

Solution:

Let V and r be the volume and diameter of the sphere, respectively. Then,

$V=\frac{4}{3} \pi(\text { radius })^{3}$

$\Rightarrow V=\frac{4}{3} \pi\left(\frac{r}{2}\right)^{3}=\frac{1}{6} \pi r^{3}$

$\Rightarrow \frac{d V}{d r}=\frac{1}{2} \pi r^{2}$

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