Answer
$r^{\prime}(x)=-\displaystyle \frac{8}{3x^{3}}+\frac{3.2}{x^{4.2}}$
Work Step by Step
SUMMARY:
The Power Rule$:\ \ \ [x^{n}]^{\prime}=nx^{n-1 } $
Sum Rule: $\ \ \ \ \ \ [f\pm g]^{\prime}(x)=f^{\prime}(x)\pm g^{\prime}(x) $
Constant Multiple Rule:$\ \ \ [cf]^{\prime}(x)=cf^{\prime}(x) $
Constant times x:$\ \ \ \displaystyle \frac{d}{dx}(cx)=c $
Constant:$\displaystyle \ \ \ \ \ \frac{d}{dx}(c)=0 $
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$r^{\prime}(x)=[ \displaystyle \frac{4}{3}x^{-2}-x^{-3.2}]^{\prime}=... $Sum Rule,
$=[\displaystyle \frac{4}{3}x^{-2}]^{\prime}-[x^{-3.2}]^{\prime}=...$Constant Multiple Rule
$=\displaystyle \frac{4}{3}[x^{-2}]^{\prime}- [x^{-3.2}]^{\prime}$=...Power Rule...
$=\displaystyle \frac{4}{3}[-2\cdot x^{-3}]- [-3.2\cdot x^{-4.2}]$
$r^{\prime}(x)=-\displaystyle \frac{8}{3x^{3}}+\frac{3.2}{x^{4.2}}$
