Answer
$t^{\prime}(x)=\displaystyle \frac{3|x|}{x}-\frac{1}{2\sqrt{x}}$
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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$t^{\prime}(x)=[3|x|-x^{1/2}]^{\prime}=... $Sum Rule,
$t^{\prime}(x)=[3|x|]^{\prime}-[x^{1/2}]^{\prime}$
$[|x|]^{\prime}=\displaystyle \frac{|x|}{x}$ (from sec.10.6)...Constant Multiple Rule, Power Rule...
$t^{\prime}(x)=3(\displaystyle \frac{|x|}{x})-\frac{1}{2}x^{-1/2}$
$t^{\prime}(x)=\displaystyle \frac{3|x|}{x}-\frac{1}{2\sqrt{x}}$
