Show a polar function's diffrentiability

Question

I need to show that $f(r,\theta)=r\sin(2\theta)\ r>0$ is differentiable at each point in its domain, and also decide whether it's $C^1$ or not. How should I approach this?

Answer 1

$\frac{\partial f}{\partial r}=\sin(2\theta)$ is continuous and $\frac{\partial f}{\partial \theta}=2r\cos(2\theta)$ which is continuous too. Then $f\in\mathcal C^1(\mathbb R^*_+\times [0,\pi[)$ and so $f$ is differentiable.