Differentiability of solution to ODE

Consider the problem
$$frac{d X(t,x)}{dt} = f(t, X(t,x))$$
$$X(0,x) = x$$
where $f:[0,T]times mathbb{R}^n to mathbb{R}^n$ and $X:[0,T]times mathbb{R}^n to mathbb{R}^n$. Assume that $f$ is Holder continuous or Lipschitz continuous. How can I prove and where can I find a reference for the fact that $X$ is differentiable and
$$nabla X$$ is the solution to
$$frac{d}{dt} nabla X(t,x) = f(t,X(t,x)) nabla X(t,x)$$
and in particular
$$nabla X = e^{int_0^Tnabla f}$$
? Do we need to assume that $f$ is differentiable of does the Lipschitz continuity suffice?

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