Here's how I understand the problem
let Φ = f (tx,ty) = t^n f(x,y)
dΦ/dt = (∂f/∂x)*(dx/dt) + (∂f/∂y)*(dy/dt) = n*t^(n-1) f(x,y)
this gives
dΦ/dt = (∂f/∂x)*x + (∂f/∂y)*y = n*t^(n-1) f(x,y)
to show this equals the proposition, substitute t = 1
(∂f/∂x)*x + (∂f/∂y)*y = n*f(x,y)
If...