so for some constant since we get

first you need to find the unit vector in this direction: we have thus the unit vector is:(a) Find the rate of change of T at (1,2,2) in the direction toward the point (2,1,3).

it's also clear that now the rate of change of is:

the direction of greatest increase at any point is the direction of at that point. but in part (a) we saw that(b) Show that at any point in the ball the direction of greatest increase in temperature is given by a vector that points toward the origin.

and have the same direction and, clearly, the vector points toward the origin. Q.E.D.