I have a flow defined by the initial value problem:
$$\frac{d}{dt}y_t(x)=f_t(y_t(x)), \quad y_0(x)=x$$
where $f_t:\mathbb{R}^k\rightarrow\mathbb{R}^k$. I know the above problem has a unique solution for $t\in[0,1]$ and $x\in\mathbb{R}^k$, and I know that $f_t$ is continuously differentiable with respect to $x$. I have seen it claimed that these facts imply that $y_t$ is continuously differentiable with respect to $x$ but I don't see how this follows... please can someone show, or at least point me to a text showing, that $y_t$ must be continuously differentiable with respect to $x$?