Find the direction in which the function is increasing most rapidly at the point Po. f(x, y, z)= xy -lnz , Po (1,1,1)

The largest rate of change occurs in the same direction as the gradient of f at the point.

∇f(x, y, z) = (∂f/∂x, ∂f/∂y, ∂f/∂z) = (y, x, -1/z)

==>   ∇f (1, 1, 1) = (1, 1, -1)

In other words, f changes at the highest rate in the direction of the vector (1, 1, -1).