Is there a proof that $L^p([0,1],X)$ has the Radon Nikodym Property that uses the characterization via Lipschitz continuous functions?

Question

I'm looking for an alternative proof of the following theorem: Let $1<p<\infty$ then $L^p([0,1],X)$ has the Radon Nikodym property iff $X$ does. The theorem is due to Sundaresan and an alternative proof was given by Turett & Uhl

I was wondering if there exists a proof of this Theorem, using this characterization: $X$ has the Radon Nikodym property iff every Lipschitz continuous function $f:[0,1]\rightarrow X$ is differentiable a.e.

Does anyone know a source for this?