Answer
$r^{\prime}(x)=-\displaystyle \frac{2}{3x^{2}}+\frac{0.1}{2x^{1.1}}$
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{2}{3}x^{-1}-\frac{1}{2}x^{-0.1}]^{\prime}=... $Sum Rule,
$=[\displaystyle \frac{2}{3}x^{-1}]^{\prime}-[\frac{1}{2}x^{-0.1}]^{\prime}=...$Constant Multiple Rule
$=\displaystyle \frac{2}{3}[x^{-1}]^{\prime}-\frac{1}{2}[x^{-0.1}]^{\prime}$=...power rule...
$=\displaystyle \frac{2}{3}[-1\cdot x^{-2}]-\frac{1}{2}[-0.1\cdot x^{-1.1}]$
$=\displaystyle \frac{2}{3}[-1\cdot x^{-2}]-\frac{1}{2}[-0.1\cdot x^{-1.1}]$
$=-\displaystyle \frac{2}{3x^{2}}+\frac{0.1}{2x^{1.1}}$
