Answer
$${x^{\sin x}}\left( {\frac{{\sin x}}{x} + \ln x\cos x} \right)$$
Work Step by Step
$$\eqalign{
& \frac{d}{{dx}}\left( {{x^{\sin x}}} \right) \cr
& {\text{set }}y = {x^{\sin x}} \cr
& {\text{taking the natural logarithm of both sides of the equation}} \cr
& \ln y = \ln {x^{\sin x}} \cr
& {\text{using logarithmic properties}} \cr
& \ln y = \sin x\ln x \cr
& {\text{differentiate both sides}} \cr
& \frac{d}{{dx}}\left[ {\ln y} \right] = \frac{d}{{dx}}\left[ {\sin x\ln x} \right] \cr
& \frac{1}{y}\frac{{dy}}{{dx}} = \sin x\frac{d}{{dx}}\left[ {\ln x} \right] + \ln x\frac{d}{{dx}}\left[ {\sin x} \right] \cr
& {\text{solving derivatives}} \cr
& \frac{1}{y}\frac{{dy}}{{dx}} = \sin x\left( {\frac{1}{x}} \right) + \ln x\cos x \cr
& \frac{{dy}}{{dx}} = y\left( {\frac{{\sin x}}{x} + \ln x\cos x} \right) \cr
& {\text{replace }}y{\text{ with }}{x^{\sin x}} \cr
& \frac{{d\left( {{x^{\sin x}}} \right)}}{{dx}} = {x^{\sin x}}\left( {\frac{{\sin x}}{x} + \ln x\cos x} \right) \cr
& \frac{d}{{dx}}\left( {{x^{\sin x}}} \right) = {x^{\sin x}}\left( {\frac{{\sin x}}{x} + \ln x\cos x} \right) \cr} $$
