directional derivatives in (0,0) I've this function : $f(x,y)= \dfrac{(1+x^2)x^2y^4}{x^4+2x^2y^4+y^8}$ for $(x,y)\ne (0,0)$ and $0$ for $(x,y)=(0,0)$
It's admits directional derivatives at the origin?

A function admits directional derivative at a point if its gradient $\nabla{f}$ exists at that point. The gradient of your function is given by,
$$\nabla{f}=\left(\begin{array}{cc} -\frac{2\, x\, y^4\, \left(2\, x^6 + 3\, x^4 - 2\, y^4 + 1\right)}{{\left(x^6 + x^2 + 2\, y^4\right)}^2} & \frac{4\, x^2\, y^3\, \left(x^6 + x^4 + x^2 + 1\right)}{{\left(x^6 + x^2 + 2\, y^4\right)}^2} \end{array}\right)$$
which doesn't exist at $(x,y)=(0,0)$, i.e. the origin, since you have $\frac{0}{0}$ which is undefined.
