Answer
$f^{\prime}(x)=0.5x^{-0.5}-x^{-1.5}$
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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$f^{\prime}(x)=[x^{0.5}+2x^{-0.5}]^{\prime}=... $Sum Rule,
$=[x^{0.5}]^{\prime}+[2x^{-0.5}]^{\prime}=$... individually:
$[x^{0.5}]^{\prime}$=...power rule...$=0.5x^{-0.5}$
$[2x^{-0.5}]^{\prime}=$...Constant Multiple Rule...
$=2[x^{-0.5}]^{\prime}$=...power rule...$=2(-0.5x^{-1.5})=-x^{-1.5}$
So
$f^{\prime}(x)=0.5x^{-0.5}-x^{-1.5}$
