Given the $2$D scalar field, $T(x,y)=x^3{y}^2+3x^{2}yln(x),$ where $ln(*)$ is the natural logarithm function.

$($a $|$Find the gradient of $T(x,y)$ in Cartesian coordinate system;

$($b$)$ Find the directional derivative del$\frac{T}{d}$els in the direction $s=$hat$(i)+2$hat$(j)$ defined in Cartesian coordinate system;

$($c$)$ Find the direction of maximum change of $T$ at the point $(2,2)$ and write your answer in terms of unit vector $($Hint: gradient$).$