The direction of steepest increase of a function f is given by its gradient, grad(f) or ∇f; which you already calculated to be (2x,3y). You don't know the exact value of the speed r'(t) but you do know its direction (the gradient), so both are multiples of eachother:
This is an easy system of two linear differential equations that you can solve combined with the initial values x(0) = y(0) = 1. The given initial speed will allow you to determine the constant of proportionality k that we introduced. Does that help?