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

$T(x,y,z)=100e^{-x^2-5y^2-7z^2}$

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

$($a$)$ Find the rate of change of temperature $($in $\frac{C}{m})$ at the point $P(4,-1,5)$ in the direction toward the point $(6,-5,6).$

$\frac{C}{m}$

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

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