Let $\displaystyle S$ be the system of equations $\displaystyle x-u-v=0,$ $\displaystyle y-u^2-v^2=0$ and $\displaystyle z-u^3-v^3=0.$ Find point $\displaystyle P=\{x_0,y_0,z_0,u_0,v_0\}$ such that:

a) Around $\displaystyle P,$ the system $\displaystyle S$ implicitly defines three differentiable functions: $\displaystyle z=f_1(x,y),\, u=f_2(x,y),\,v=f_3(x,y).$

b) The directional derivative of $\displaystyle f_1$ in $\displaystyle (x_0,y_0)$ is maximum according the direction of the vector $\displaystyle (0,1)$ and the value of the aforesaid directional derivative is $\displaystyle 3.$


I don't get very well what to do in part a), as for b), how can I compute the directional derivative?