6.

(a) (5%) Give the definition of the directional derivative of $f(x,y)$ at $(a,b)$ in the direction of a unit vector $\mathbf{u}=(u,v)$, $u^2+v^2=1$.

(b) (5%) Show that the directional derivative of the following function exists in every direction $\mathbf{u}=(u,v)$ at $(0,0)$, where

$$f(x,y)= \begin{cases} \dfrac{xy^2}{x^2+y^4}, & \text{if }(x,y)\ne (0,0);\\ 0, & \text{if }(x,y)=(0,0). \end{cases}$$

(c) (5%) Show that the above function $f(x,y)$ is not differentiable at $(0,0)$.