Analysis

Given

Find

$\stackrel{\.}{S}\left(\stackrel{\^}{t}\right)$

Define the functions $V\left(t\right)$ and $S\left(t\right)$ 

$V\left(t\right)\=\frac{4}{3} \mathrm{\π} r^3\left(t\right)$$\stackrel{\text{assign as function}}{\to }$$V$

$S\left(t\right)\=4 \mathrm{\π} r^2\left(t\right)$$\stackrel{\text{assign as function}}{\to }$$S$

Solve the equation $\stackrel{\.}{V}\left(t\right)\=25$ for $\stackrel{\.}{r}$

$\stackrel{\.}{V}\left(t\right)\=25$

$4\mathrm{\π}{r\left(t\right)}^2\left(\frac{\ⅆ}{\ⅆt}r\left(t\right)\right)\=25$

$\stackrel{\text{isolate for diff(r(t),t)}}{\to }$

In $\stackrel{\.}{S}\left(t\right)$, replace $\stackrel{\.}{r}$ with its value from the equation $\stackrel{\.}{V}\left(t\right)\=25$ 

$\genfrac{}{}{0ex}{}{\stackrel{\.}{S}\left(t\right)}{\phantom{x\=a}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{f(x)}}}{}$ = $\frac{50}{r\left(t\right)}$$$