Let $f\in C^2(\mathbb{R}^2,\mathbb{R})$, $g(\rho,\theta)=(\rho\cos\theta,\rho\sin\theta)$ with $\rho\gt 0$ and $\theta \in [0,2\pi)$. Let $h=f\circ g$, we calculate: $$\nabla h(\rho,\theta)=\nabla f(g(\rho,\theta))\nabla g(\rho,\theta)$$ Getting $\nabla h(\rho,\theta)=(\cos\theta f_x(\rho \cos\theta,\rho \sin\theta)+\sin\theta f_y(\rho \cos\theta,\rho \sin\theta),-\rho\sin\theta f_x(\rho \cos\theta,\rho \sin\theta)+\rho\cos\theta f_y(\rho \cos\theta,\rho \sin\theta))=(h_\rho(\rho,\theta),h_\theta(\rho,\theta))$
What I'm not sure about is how can I calculate the second partial derivatives, say I wanted to calculate $h_{\rho\rho}(\rho,\theta)$, if I try to derive $h_\rho$ with respect to $\rho$ I believe I would have: $$h_{\rho\rho}=\cos\theta(f_x(\rho\cos\theta,\rho\sin\theta))_\rho+\sin\theta(f_y(\rho\cos\theta,\rho\sin\theta))_\rho$$ But how do I calculate $(f_x(\rho\cos\theta,\rho\sin\theta))_\rho$ and $ (f_y(\rho\cos\theta,\rho\sin\theta))_\rho$? (Or how should I calculate $h_{\rho\rho}$?)
If I was interessed in calculating $\nabla^2 h(\rho,\theta)$ would this be the best approach? just computing the $4$ second dervatives?