$\displaystyle f(x,y)=\sqrt[5]{x^5+y^5}$

The question is to find $\displaystyle f_x(0,0)$ and $\displaystyle f_{xx}(0,0)$

Solving $\displaystyle f_x(0,0)$ how I would normally take a partial derivative I get $\displaystyle \frac{x^4}{(x^5+y^5)^{\frac{4}{5}}}$ which is undefined at (0,0).

I get 1 when I use the limit definition of the partial derivative. I'm unsure how to calculate the the second partial though. I want to get an actual number and not 0/0 if possible.