Problem

A curve $C$ has equation

$$y=x^{\sin x}\qquad x>0\qquad y>0$$

(a) Find, by firstly taking natural logarithms, an expression for $\dfrac{dy}{dx}$ in terms of $x$ and $y$.

(b) Hence show that the $x$ coordinates of the stationary points of $C$ are solutions of the equation

$$\tan x+x\ln x=0$$