Question

Let g be a differentiable function of one variable with g(2) = 2, g'(2) = -1. In what direction at the point (-1,-2) does the function f(x,y) =xg(y/x) increase most rapidly?

Answer 1

In the direction of gradient

Answer 2

take partial derivatic of f w.r.t x=g(y/x)+xg'(y/x)(-y/x^2)=g(y/x)-y/x g'(y/x)
take partial derivative of w.r.t y=g'(y/x)
so grad(f)=[g(y/x)+xg'(y/x)(-y/x^2)=g(y/x)-y/x g'(y/x)]i+[g'(y/x)]j
at point (-1,-2)
grad(f)=[g(2)-2g'(2)]i+g'(2)j
=[2+2]i-j=4i-j