Find the directional derivative of $$ f(x,y) := \begin{cases} \frac{xy^2}{x^2+y^4}, & \text{for } (x,y)\neq (0,0) \newline 0, & \text{otherwise.} \end{cases} $$ at $(0,0)$ in direction of vector $u=\langle \sqrt{2},\sqrt{2}\rangle$.
Since it is a piecewise defined function I want to make use of the formula: $$ D_uf(x,y) =\lim_{h\to 0}\frac{f(x+hu_x,y+hu_y)-f(x,y)}{h}, $$ where $D_uf(x,y)$ is the directional derivative of $f$ in direction of vector $u=\langle u_x,u_y \rangle$ at $(x,y)$.
My only concern here is: do I need to normalise the vector $u$ before applying this formula?
For my question, is $u_x=u_y=\sqrt{2}$ or $u_x=u_y=\frac{1}{\sqrt{2}}$ (obtained after normalising the vector $u$)?
You can even take the vector $\;\langle 1,1\rangle\;$ for an easier computation, it has same direction.
But if you do not normalize before, you have to divide by the norm of the vector after. The order doesn't play a role.