Conformal Map

F : R^2 -> R^2, (x,y) -> (u(x,y) , v(x,y))
u and v have continous partial u_x , u_y ,v_x, v_y
u_x = partial of u w.r.t. x
where u_x*v_y -  u_y*v_x not equal to 0
(x0,t0) at t = 0
consider
a1(t) =(x0+t , y0)
a2(t) =(x0, y0+t)
where a1(t) is perpendicular to a2(t)

let A1(t) = F(a1(t))
A2(t) = F(a2(t))

at t = 0
show that
derivative of (A1(t)) dot product with derivative of (A2(t))  = u_x*u_y -  v_y*v_x

and if F preserve angle then u_x*u_y -  v_y*v_x = 0

can anyone help ??