Let $T_v$ be a translation by $v$ (i.e. $T_v(x,y)=(x+v_x,y+v_y)$).
Let $f:\mathbb{R}^2\to\mathbb{R}$ be a smooth function (i.e. have derivatives of all orders).
We define the following differential operator : $P_x(f)(v)=(\frac{d}{dt}(f(T_{(t,0)}v))(0)$ (evaluated at t=0).
I don't know how to derive this expression, can someone please help ? (The translations should be with a 1x2 vector and not 2x1, but I don't know this notation in tex).