Consider the function $f$:$R^{2}R,$ with

$f(x,y)=e^{x*y}{x}^{2}y-2y,(x,y)in{R}^2$

$($a$)$ Show that the gradient gradf of $f$ equals

gradf$(x,y)=(y{e}^{x*y}2xy,x{e}^{x*y}{x}^2-2),(x,y)in{R}^2$

$[5$ marks$]$

$($b$)$ Find the tangent plane of its graph at the point $A(0,-1,3).$

$[10$ marks$]$

$($c$)$ Find the derivative $D_{u}f(0,-1)$ of $f$ in the direction of the vector $u=(3,-4)$ at the point $(0,-1).$

$[10$ marks$]$

$($d$)$ Consider the composite function $f(t)=f(x(t),y(t)),$ where $x(t)=2t$ and $y(t)=-3t,$ for tinR. Use the chain rule to compute $f^{\text{'}}(t).$

$[10$ marks$]$