The temperature at a point $($x$,$ y$,$ z$)$ is given by T$($x$,$ y$,$ z$)=200$e$$x$^25$y$^29$z$^2$ where T is measured in $$C and x$,$ y$,$ z in meters.

$($a$)$ Find the rate of change of temperature at the point P$(2,1,5)$ in the direction towards the point $(2,1,5).$C$/$m

$($b$)$ In which direction does the temperature increase fastest at P$?$

$($c$)$ Find the maximum rate of increase at P$.$