Answer
$\frac{1}{2x\ln 10}$
Work Step by Step
$\frac{d}{dx}\log_{10}\sqrt{x}$
$=\frac{1}{\sqrt{x}\ln 10}\frac{d}{dx}\sqrt{x}$
$=\frac{1}{\sqrt{x}\ln 10}\frac{1}{2}x^{-\frac{1}{2}}$
$=\frac{1}{\sqrt{x}\ln 10}\frac{1}{2\sqrt{x}}$
$=\frac{1}{2x\ln 10}$