calculus of two variables question?

Question

say heat flow is being modelled by the temperature in the xy-plane is given by:
T(x,y)=x*y - x

I've found the general directional derivative at the point (2,2), and the direction that temperature changes most rapidly from there, and now i'm being asked this:

If heat flow is a direction -∇T, express this as a differential equation in y and x and solve it. I'm sure I could solve it but I don't understand how you find the differential equation. how do you express it as one?

please help!

by the way where i've written " -∇T "
that's meant to be a nabla - one of those upside down triangles, it didn't seem to come out very well though...

Answer 1

If you have the function, you take the two partials
d(partial) T/dx and d(partial)T/dy. Where x is the variable, y is treated as a constant, and vice versa. The negative sum is -delta T.