is the function $f$ differentiable at $(0,0)$?

To show that $f$ is differentiable at $(0,0)$ you have to show that $$f(h)=f(0,0)+\nabla f (0,0) \cdot h + o(|h|)$$ for $h \in \Bbb{R}^2$ in a neighbourhood of $(0,0)$ (here $\cdot$ denotes the scalar product). It is natural to put $\nabla f (0,0) = (0,0)$, so that indeed you need to prove $$\lim_{h \to (0,0)} \frac{f(h)-f(0,0)}{|h|} =0$$ Using polar coordinates $h=(R \cos \theta , R \sin \theta)$ you have $$0 \le \frac{f(h)-f(0,0)}{|h|} = \frac{R^4 \cos^2 \theta \sin^2 \theta}{R^2} \le R^2 \to 0$$ and you are done.

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