doubt about implicit function theorem problem

Question

I have a doubt.

Let $\omega=]0, +\infty[$ and $g:\omega \rightarrow \mathbb{R}$ defined by $g(x,y)=xycos(xy)$

I want to prove that $\forall$ $(x_0, y_0) \in \omega$ for whom $g(x,y) +\pi =0$ we can define x has a function of y such as $y=\phi(x) $.

The problem is... How can i prove that $\frac{\partial g} {\partial y} \neq 0, \forall (x_0,y_0) $?

Best regards