For the following functions $f$ and points $P,$ answer the following.

$($a$)$ Find the directional derivative $D_{u}f$ at $P$ in the direction of $v.$

$($b$)$ Find the direction in which $f$ increases most rapidly at $P.$ Find the value of the directional derivative in this direction.

$($c$)$ Find the equation of the tangent space to $f$ at $P.$

$($i$)f(x,y)=e^{x}cosy,P(0,0),v=($:$2,3$:$)$

$($ii$)f(x,y,z)=arctan(xyz),P(1,1,0)*v=($:$1,2,3$:$)$

$($iii$)f(x,y,z)=\sqrt[\phantom{2}]{x^2+y^2+z^2},P(1,2,2).v=($:$1,1,1$:$)$