How to show $d(x_0,\exp_x(v))$ is smooth? $(M,g)$ is a smooth Riemann manifold.$d(\cdot,\cdot)$ is distance function on $(M,g)$. $x_0$ is a given point in $M$. $\exp$ is exponent map.
If $v\in T_xM$ is enough small, how to show $d(x_0,\exp_x(v))$ is smooth about $x$? 
I mean that $d(x_0,\exp_x(v))\in C^\infty (M)$  ? In fact ,I think the question is show that $\exp_x(v)$  is smooth about $x$. Seemly, it's a little strange.Because I think for different $x_1$ and $x_2$, there are nothing between $\exp_{x_1}$ and $\exp_{x_2}$.
 A: As  in some other question from yesterday (you seem to be using two accounts?, lanse7pty and lanse2pty? Anyway, that's your decision) $d(x_0, \exp_x(v))$ is not a function of $x$ but a function of $x$ and $v, v\in T_xv$. Therefore it does not make sense to say this is smooth about $x$. Either it is smooth as a function of $x$ and $v$, or you have to fix a vector field $v$ in a neighbourhood or $x$ to view this as a function of $x$ alone. 
And yes, the first function is knwown to be smooth if you stay away from the diagonal in the distance function (you can square it to make it smooth). This is a consequence of the theorem about smooth dependence of solutions of ordinary differential equations from the initial conditions and the local differentiability of the Riemannian distance function. 
If the theorem  about smmoth dependence of solutions of differential equations is known to you (a proof should be readily available in textbooks on ODE. One proof known to me can be found in one of Serge Langs books, I think it was 'Real Analysis' (I cannot check right now, sorry)), then a proof of how to show $(x,v)\mapsto \exp_xv$ is found in (probably any book on differential geometry, e.g. in) Klingenberg, Riemannian Geometry (de Gruyter), 2nd edition, Proposition 1.6.10.
Finally, local smoothness of the distance function (if you stay away from the diagonal) is a consequence of the so called Gauss Lemma (which states that the exponential map, restricted to some tangent space $T_vM$, is a radial isometry. This can also be found in Klingenbergs book (or any other textbook on Riemanannian Geometry)
Edit: when saying 'local' then I mean that $x\mapsto d(x_0,x )$ is smooth in a neighbourhood $V=U(x_0)\backslash\{x_0\}$ with $x_0$ deleted. It is not globally smooth, but only as long as geodesics emanating from $x_0$ are unique length minimizers. Similar statements are true if you look at the distance function to smooth submanifolds.
