# What is the derivative of f(x)=sqrt(1+log_3(x) ?

$\frac{d}{\mathrm{dx}} \left(\sqrt{1 + {\log}_{3} x}\right)$
$= \frac{\left(\frac{d}{\mathrm{dx}}\right) \left(1 + {\log}_{3} x\right)}{2 \sqrt{1 + {\log}_{3} x}}$
$= \frac{\left(\frac{d}{\mathrm{dx}}\right) \left(1 + \log \frac{x}{\log} 3\right)}{2 \sqrt{1 + {\log}_{3} x}}$
$= \frac{\frac{1}{x \ln 3}}{2 \sqrt{1 + {\log}_{3} x}}$
$= \frac{1}{2 x \ln 3 \sqrt{1 + {\log}_{3}}}$