Checking for differentiability,partial derivatives of a multivariable function

Let $f : \mathbb{R}^2 \to \mathbb{R}$ be defined as

$$f(x,y)=\begin{cases} (x^2+y^2)\cos \frac{1}{\sqrt {x^2+y^2}}, & \text{for $(x,y) \ne (0,0)$} \\ 0, & \text{for $(x,y) = (0,0)$} \end{cases}$$
then check whether its differentiable and also whether its partial derivatives ie $f_x,f_y$ are continuous at $(0,0)$. I dont know how to check the differentiability of a multivariable function as I am just beginning to learn it. For continuity of partial derivative I just checked for $f_x$ as function is symmetric in$y$and $x$. So $f_x$ turns out to be $$f_x(x,y) = 2x\cos \left(\frac {1}{\sqrt {x^2+y^2}}\right)+\frac {x}{\sqrt {x^2+y^2}}\sin \left(\frac {1}{\sqrt{x^2+y^2}}\right)$$ which is definitely not $0$ as $(x,y)\to (0,0)$. Same can be stated for $f_y$. But how to proceed with the first part? Thanks!