Hello,

Working on the following problem:

Express the rate at which the volume of the sphere changes with respect to the surface area of the sphere (as a function of r).

$\displaystyle S = 4\pi r^2 \Rightarrow r = \left( \frac{S}{4\pi}\right)^{\frac{1}{2}}$

My Work:

$\displaystyle V = \frac{4}{3} \pi r^3 = \frac{4}{3}\pi \cdot \left( \frac{S}{4\pi}\right)^{\frac{3}{2}}$

$\displaystyle \frac{d}{dS} \left( V = \frac{4}{3} \pi \cdot \left( \frac{S}{4\pi}\right)^{\frac{3}{2}\right)$

$\displaystyle \frac{dV}{dS} = \frac{1}{2} \left(\frac{S}{4\pi}\right)^{\frac{1}{3}}$

$\displaystyle \text{Substitute } S:$

$\displaystyle \frac{dV}{dS} = \frac{1}{2} \left( \frac{4\pi r^2}{4\pi}\right)^{\frac{1}{3}} = \frac{1}{2} (r^2)^{\frac{1}{3}} = \frac{1}{2} \cdot r^{\frac{2}{3}}$

My brain is fried after doing a lot of calculus problems so I wanted a sanity check: does it look okay?