Answer
$ \displaystyle \frac{dA}{dr}=8\pi r$
Work Step by Step
SUMMARY (rules in differential notation):
1. The Power Rule$:\ \ \ \displaystyle \frac{d}{dx}[x^{n}]=n\cdot x^{n-1 } $
2. Sum Rule: $\displaystyle \ \ \ \frac{d}{dx}[f\pm g](x)=\frac{d}{dx}[f(x)]\pm\frac{d}{dx}[g(x)] $
3. Constant Multiple Rule:$\ \ \displaystyle \frac{d}{dx}[cf(x)]=c\cdot\frac{d}{dx}[f(x)] $
4. Constant times x:$\ \ \ \displaystyle \frac{d}{dx}(cx)=c $
5. Constant:$\displaystyle \ \ \ \ \ \frac{d}{dx}(c)=0 $
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$ \displaystyle \frac{dA}{dr}= \frac{d}{dr}[4\pi r^{2}]$= $\ \ \ $...($ 3, \pi$ is a constant too)
$=4\displaystyle \pi\cdot\frac{d}{dr}[r^{2}]$= $\ \ \ $...($1$)
$=4\pi(2r)$
$ \displaystyle \frac{dA}{dr}=8\pi r$
