Following my previews question : From the comments to the answer I feel that I don't understand how to use the chain rule.
From what I understand the chain rule sais : $F(t)=f(x(t),y(t))\implies F'(t)=\langle(\frac{\partial f}{\partial x},\frac{\partial f}{\partial y}),(x'(t),y'(t))\rangle$. [with the right conditions]
Here $F(t)=f(x(t),y(t))$ where $x(t)=x+t,y(t)=y\implies x'(t)=1,y'(t)=0\implies{P_{x}(f)(v)=(\frac{d}{dt}(f(T_{\begin{pmatrix}t\\ 0 \end{pmatrix}}v))|_{t=0}=(\frac{\partial f}{\partial x}1+\frac{\partial f}{\partial y}0)(v)=\frac{\partial f}{\partial x}v}$.
So I concluded that the fact we evaluate at $t=0$ is irrelevant (that is evaluating at a different point will not change the answer), but I was told I am wrong (in the comments).
Am I using the chain rule wrong ?