$f:\mathbb{R}^2 \to \mathbb{R}$
$f\Bigg(\begin{matrix}x\\y\end{matrix}\Bigg)=\begin{cases}\frac{xy^2}{x^2+y^2},(x,y)^T \neq(0,0)^T \\0 , (x,y)^T=(0,0)^T\end{cases}$
I need to determine all partial derivatives for $(x,y)^T \in \mathbb{R}^2$:
$f_x=y^2/(x^2+y^2)-2x^2y^2/(x^2+y^2)^2$ for $(x,y)^T \neq (0,0)$
$f_y=2xy/(x^2+y^2)-2xy^3/(x^2+y^2)^2$ for $(x,y)^T \neq (0,0)$
and $f_x=f_y=0$ for $(x,y)^T = (0,0)$.
Then I need to determine $\frac{\partial f}{\partial v}((0,0)^T)$ for all $v=(v_1,v_2)^T \in \mathbb{R}^2$.
I tried: $\frac{1}{s} (f(x+sv)-f(x))$ at $x=(0,0)^T$ is equal to $\frac{1}{s} f(sv)$=$\frac{1}{s} f\Big(\begin{matrix}sv_1\\sv_2\end{matrix}\Big)$.
Which is either equal to $0$ when the argument is $(0,0)^T$ or it is $\frac{1}{s}\frac{sv_1s^2v_2^2}{s^2v_1^2+s^2v_2^2}$ which converges to $\frac{v_1v_2^2}{v_1^2+v_2^2}$ as $s \to \infty$.
Is that correct so far?
And how do I know if $f$ is continuously partial differentiable on $\mathbb{R}^2$? According to our professor $f$ is not differentiable at $0$. How do I show that? As far as I know it has something to do with that something is not linear but I don't know what exactly. So I guess it can't be continuously partial differentiable on $\mathbb{R}^2$ as well but I am not sure about that.
Thanks for your help!
Best Answer
The partial derivatives at $(0,0)$ are
$$\partial_x f(0,0) = \lim_{h\to 0} \frac{f(h,0) - f(0,0)}h = 0$$ $$\partial_y f(0,0) = \lim_{h\to 0} \frac{f(0,h) - f(0,0)}h = 0$$
so the only candidate for the differential $Df(0,0)$ is the zero operator.
However, the limit
$$\lim_{(h_1, h_2) \to (0,0)} \frac{\|f(h_1, h_2) - f(0,0) - Df(0,0)(h_1, h_2)\|}{\|(h_1, h_2)\|} = \lim_{(h_1, h_2) \to (0,0)} \frac{\|f(h_1, h_2)\|}{\sqrt{h_1^2 + h_2^2}} = \lim_{(h_1, h_2) \to (0,0)} \frac{h_1h_2^2}{(h_1^2 + h_2^2)^{3/2}}$$ does not exist because e.g. for $h_1 = h_2$ we get $$\lim_{h_1 \to 0} \frac{h_1^3}{|h_1|^3}$$
which is $\pm 1$, depending on whether $h_1$ approaches $0$ from the left or from the right.
Therefore, $f$ is not differentiable at $(0,0)$.