The temperature at a point $(x,y,z)$ is given by

$T(x,y,z)=200e^{-x^2-3y^2-7z^2}$

where $T$ is measured in $C$ and $x,y,z$ in meters.

$($a$)$ Find the rate of change of temperature at the point $P(2,-1,3)$ in the direction towards the point $(6,-3,5).$

$\frac{-1600e^{-4}-48}{\sqrt[\phantom{2}]{6}}$

$\frac{C}{m}$

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

$($:$-800e^{-4},6,-42$:$)$

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