Past exam of the mathematical methods II course of the University of Oxford / 2016 resit / 1 / b / Solution

Implicit differentiation gives $2y^{\prime }=15x^2+(\cos y)y^{\prime }$. ThereforeFor real $y$, the denominator is at least one, so division introduces no singularity. In fact the function $2y-\sin y$ is strictly increasing, consistent with a unique real branch $y(x)$.