Existence of partial derivatives $$f(x,y) = \left\{\begin{matrix}
\frac{x^2}{x^2 + y^2} &, (x,y) \neq (0,0) \\ 
 0&, (x,y) = (0,0)
\end{matrix}\right.
$$
Questions:


*

*$f_x(x,y)$

*$f_y(x,y)$

*$f_{yx}(0,0)$


How can i solve this question?
 A: I'm guessing that you want to calculate those quantities. For points besides $(0,0),$ there's nothing unusual to consider. At $(0,0)$, you'll need the limit definition of the partial derivative. For example, in number 1, look at
$\lim_{h\to 0} \frac{\frac{(x+h)^2}{(x+h)^2+0^2}-0}{h}.$
The second derivative will also require the limit definition at the origin. Let me know if you figure out the first two, and we'll go from there.
A: Well, for (1), you'll treat $y$ like it's a constant, then take the derivative with respect to $x$ (applying quotient rule) to get $f_x(x,y)$ for $(x,y)\neq(0,0)$. Also, $$f_x(0,0)=\lim_{h\to 0}\frac{f(h,0)-f(0,0)}{h}$$ is undefined, so $f_x$ is undefined everywhere but at the origin.
We'll do something similar for (2), but this time we treat $x$ as a constant and take the derivative with respect to $y$ to find $f_y(x,y)$ for $(x,y)\neq(0,0)$, then use $$f_y(0,0)=\lim_{k\to 0}\frac{f(0,k)-f(0,0)}{k},$$ which actually does exist, so $f_y$ is defined everywhere. (Hover over the blank spot below to see how $f_y$ is defined as a piecewise function.)

$$f_y(x,y)=\begin{cases}-\dfrac{2x^2y}{(x^2+y^2)^2} & (x,y)\neq(0,0)\\0 & (x,y)=(0,0)\end{cases}$$

For (3), we'll use our answer from (2), and proceed in the same way we did in (1), treating $y$ as a constant and taking the derivative of $f_y(x,y)$ with respect to $x$ to get $f_{yx}(x,y)$ for $(x,y)\neq(0,0)$, then finding $$f_{yx}(0,0)=\lim_{h\to 0}\frac{f_y(h,0)-f_y(0,0)}{h}.$$ 
A: 1) $f_x(x,y)$ just means the partial derivative w.r.t $x$
$$\frac{\partial f}{\partial x}=-\frac{2xy^2}{(x^2+y^2)^2}$$
2) $f_y(x,y)$ is then the partil derivative w.r.t $y$
$$\frac{\partial f}{\partial y}=-\frac{2yx^2}{(x^2+y^2)^2}$$
3) $f_{yx}$ is the first partial derivative w.r.t $y$, and then w.r.t $x$. But now we are evaluating this derivative in $(0,0)$. Can you come up with the derivative? 
