Suppose $M$ is a smooth manifold, and you're given two points $x,y$, and associated tangent vectors $v$ and $w$. I'm curious, does there exist a diffeomorphism $F$ such that $F(x)=y$ and $dF_x(v)=w$?
I was playing around in $\mathbb{R}^n$. I tried the translation map $F(z)=z-x+y$, so $F(x)=y$. I let $\gamma:I\to\mathbb{R}^n$ be the curve defined by $\gamma(t)=x+tv$, but it turned out that $$ dF_x(v)=dF_x(\gamma'(0))=(F\circ\gamma)'(0)=v $$
Is it possible to make this work, hopefully for an arbitrary smooth manifold $M$? I don't mind assuming $M$ is connected, if that turns out to be a necessary hypothesis.