Answer
$r^{\prime}(x)=\displaystyle \frac{2}{3}+\frac{0.1}{2x^{0.9}}-\frac{4.4}{3x^{2.1}}$
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{2x}{3} -\displaystyle \frac{x^{0.1}}{2}+\frac{4}{3}x^{-1.1}-2]^{\prime}=... $Sum Rule,
$=[ \displaystyle \frac{2x}{3}]^{\prime} -[\displaystyle \frac{x^{0.1}}{2}]^{\prime}+[\frac{4}{3}x^{-1.1}]^{\prime}-[2]^{\prime}=...$Constant Multiple Rule
$=\displaystyle \frac{2}{3}[ x^{1}]^{\prime} -\displaystyle \frac{1}{2}[x^{0.1}]^{\prime}+\frac{4}{3}[x^{-1.1}]^{\prime}-2[x^{0}]^{\prime}=$...Power Rule...
$=\displaystyle \frac{2}{3}[ 1x^{0}] -\displaystyle \frac{1}{2}[-0.1x^{-0.9}]+\frac{4}{3}[-1.1x^{-2.1}]-2[0]^{\prime}$
$r^{\prime}(x)=\displaystyle \frac{2}{3}+\frac{0.1}{2x^{0.9}}-\frac{4.4}{3x^{2.1}}$
