The Chain Rule & Implicit Differentiation · Challenge problem

Near the point $(2, 4)$, the equation $x^y = y^x$ (with $x, y > 0$) defines $y$ as a differentiable function of $x$ along the branch through $(2,4)$. Find $\dfrac{dy}{dx}$ at $(2, 4)$. Give your answer as a decimal rounded to $4$ decimal places.