The temperature at a point $(x,y,z)$ is given by $T(x,y,z)=200e^{-x^2-\frac{y^2}{4}-\frac{z^2}{9}},$ where $T$ is measured in degrees celsius and $x,y,$ and $z$ in meters. There are lots of places to make silly errors in this problem; just try to keep track of what needs to be a unit vector.

Find the rate of change of the temperature at the point $(-1,-1,-1)$ in the direction toward the point $(-2,3,-2).$

In which direction $($unit vector$)$ does the temperature increase the fastest at $(-1,-1,-1?$

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What is the maximum rate of increase of $T$ at $(-1,-1,-1)?$