When is an SDE solution differentiable in its starting value?

Question

Let $W=\{W_t\}_{t\in[0;T]}$ be a real-valued Brownian motion, $\{F_t\}_{t\in [0;T]}$ the filtration generated by $W$, augmented with the nullsets, $\mu,\sigma\colon [0;T]\times\mathbb R \to \mathbb R$ measurable functions with some proper conditions such that, for all $t \in [0;T]$ and $x \in \mathbb R$, there is a unique stochastic process $\{X^{t,x}_s\}_{s\in[t;T]}$ satisfying $$ \mathrm d X^{t,x}_s = \mu(s,X^{t,x}_s) \mathrm ds + \sigma(s,X^{t,x}_s) \mathrm dW_s, \quad X^{t,x}_t = x. $$ If we fix $s \in (t;T]$ and a random outcome $\omega$, under which stronger conditions can we say that $x \mapsto X^{t,x}_s(\omega)$ is differentiable? Is there literature on this question? Thank you!