Answer
$$\frac{{dy}}{{dx}} = \frac{3}{{\left( {\ln 4} \right)x}}$$
Work Step by Step
$$\eqalign{
& y = {\log _4}x + {\log _4}{x^2} \cr
& {\text{use the logarithmic property }}{\log _a}{b^n} = n{\log _a}b \cr
& y = {\log _4}x + 2{\log _4}x \cr
& {\text{Find the derivative of }}y{\text{ with respect to }}x \cr
& \frac{{dy}}{{dx}} = \frac{d}{{dx}}\left[ {{{\log }_4}x} \right] + 2\frac{d}{{dx}}\left[ {{{\log }_4}x} \right] \cr
& {\text{use the rule }}\frac{d}{{dx}}\left[ {{{\log }_a}x} \right] = \frac{1}{{\left( {\ln a} \right)x}}.{\text{ Then}} \cr
& \frac{{dy}}{{dx}} = \frac{1}{{\left( {\ln 4} \right)x}} + 2\left( {\frac{1}{{\left( {\ln 4} \right)x}}} \right) \cr
& {\text{simplifying}} \cr
& \frac{{dy}}{{dx}} = \frac{3}{{\left( {\ln 4} \right)x}} \cr} $$
