## System of equations

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

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

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

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