Differentiability of solution to ODE

Question

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 $$\frac{d}{dt} \nabla X(t,x) = \nabla f(t,X(t,x)) \nabla X(t,x)?$$ Do we need to assume that $f$ is differentiable of does the Lipschitz continuity suffice? Also, does the formula $$\nabla X = e^{\int_0^T\nabla f}$$ hold if at least $f \in L^1((0,T),W^{1,1}(\mathbb{R}^n))$ for example? If not, why?