Suppose one has a smooth curve $\gamma:\mathbb{R}\to M$ on a manifold $M$ and a smooth one-parameter family $f_t:M\to M$ of diffeomorphisms (not necessarily a one-parameter subgroup). Then, the curve $\rho(t)=f_t(\gamma(t))$ is smooth.
My question is: is there a way to relate the velocity of $\gamma$ with the one of $\rho$? How may one derive $\frac{d}{dt}f_t(\gamma(t))$?