<s>[INST] Check if $\frac{x^3+y^3}{|x|+|y|}$ is differentiable at $(0,0)$ We have the function $$f(x,y) = \begin{cases} \frac{x^3+y^3}{|x|+|y|},& (x, y) \ne (0,0) \\
0, &   (x,y) = (0,0)
\end{cases}$$
My work so far:
I tried checking if $f$ is continuous at $(0,0)$:
When $x >0, y>0:$
$$\lim\limits_{(x,y)\rightarrow(0,0)} f(x) = \frac{(x+y)(x^2-xy+y^2)}{x+y} = 0$$
When $x >0, y<0:$ we can rewrite $y$ as $-y$, with $y>0$
$\lim\limits_{(x,y)\rightarrow(0,0)} f(x) = \frac{x^3-y^3}{x+y}$
Which I can not solve.
I calculated the partial derivatives:
\begin{align*}\frac{\partial{f}}{\partial{x}}(0,0)&=\lim\limits_{x\to 0} \frac{f(x,0)-f(0,0)}{x} =\lim\limits_{x\to 0}\frac{x^3}{|x|}=\lim\limits_{x\to 0} x^2 \mathrm{sgn}(x)= 0\\
\frac{\partial{f}}{\partial{x}}(x,y)&=\frac{3x^2(|x|+|y|)-(x^3+y^3)\mathrm{sgn}(x)}{(|x|+|y|)^2}
\end{align*} 
and analogously for $y$.
I tried checking if the partial derivatives are continuous and failed.
I tried with the definition:
$$\lim\limits_{(x,y)\rightarrow(0,0)} \frac{f(x,y)-f(0,0)-T(0,0)}{||(x,y)||} = 0,$$
 where $T(x,y) = 0$.
we get $$\lim\limits_{(x,y)\rightarrow(0,0)} \frac{x^3+y^3}{(|x|+|y|)\sqrt{x^2+y^2}}$$
And I got stuck here.
 [/INST] Yes, the partial derivatives exist at $(0,0)$ and they are both equal to $0$. Therefore either $f$ is not differentiable at $(0,0)$, or it is differentibale there and its derivative there is the null function.
And what actually occurs is that $f'(0,0)$ is the null function. In fact, if $\varphi\colon\mathbb R^2\longrightarrow\mathbb R$ is the null function, then\begin{align}f'(0,0)=\varphi&\iff\lim_{(x,y)\to(0,0)}\frac{\bigl\lvert f(x,y)-f(0,0)-\varphi(x-0,y-0)\bigr\rvert}{\bigl\lVert(x,y)\bigr\rVert}=0\\&\iff\lim_{(x,y)\to(0,0)}\frac{\lvert x^3+y^3\rvert}{\bigl(\lvert x\rvert+\lvert y\rvert\bigr)\sqrt{x^2+y^2}}=0.\end{align}But, if $x=r\cos\theta$ and $y=r\sin\theta$, then$$\lvert x^3+y^3\rvert\leqslant2r^3$$and$$\bigl(\lvert x\rvert+\lvert y\rvert\bigr)\sqrt{x^2+y^2}\geqslant r^2,$$since $\lvert\cos\theta\rvert+\lvert\sin\theta\rvert\geqslant\cos^2\theta+\sin^2\theta=1$. Therefore,$$\frac{\lvert x^3+y^3\rvert}{\bigl(\lvert x\rvert+\lvert y\rvert\bigr)\sqrt{x^2+y^2}}\leqslant r\to_{r\to0}0.$$
</s>