So x = r cos t, y = r sin t and then g(r,w) = f(x,y){under change) = f(r cos t, r sin t), so:

dg/dr = (df/dx)(dx/dr) + (df/dy)(dy/dr)= (df/dx)cos t + (df/dy)sin t

dg/dt = (df/dx)(-r sin t) + (df/dy)(rcos t)

Now just square above, add both lines and make some order in that mess and you'll get what you want, both (i) and (ii)

Tonio