Let $M$ be a smooth manifold and $(g(t))_{t\in[a,b]}$ be a family of complete Riemannian metrics (on $M$) that smoothly depend on $t$.
If $x$ and $y$ are two arbitrary points on $M$, we can define a distance function $f(t):=d_{g(t)}(x,y)$ that varies through time. The distance $d_{g(t)}$ is the usual distance induced by the Riemannian metric $g(t)$.
What I am interested in is the regularity of the function $f$, and specifically I want to use the Rademacher's theorem which guarantees me that $f$ is almost everywhere differentiable on $M$ if I can prove that $f$ is Lipschitz continuous or at least locally Lipschitz continuous.
A first idea is to use the completeness of $M$ to deduce that there is a minimising unit-speed geodesic $\gamma_t$ between $x$ and $y$ at any time, so we write : $$d_{g(t)}(x,y)=L_{g(t)}(\gamma_t)$$ But I can't prove why the family of minimising geodesic $(\gamma_t)$ would also depend smoothly on time. Moreover, I want to also prove an explicit formula for the derivative (when $f$ is differentiable ) which is : $$\frac{d}{dt}|_{t_0}d_{g(t)}(x,y)=L_{g(t_0)}(\gamma_{t_0})$$ Any help is appreciated.