Question regarding Ito Process

Question regarding Ito Process

I am new to Ito Process, so I have a following question. Consider a
standard Ito Process, $$X_t=X_0+\int_0^t\mu_sds+\int_0^t\sigma_sdW_s$$
where W is the m-dimentional Brownian motion and X is a n-dimentional
process. $\mu$ and $\sigma$ are adaptive to {$\Sigma_t$} generated by the
Brownian motion. In somewhere in my textbook, it says the drift and the
variance can be derived as following, $$\frac{d}{ds}
E(X_s|\Sigma_t)|_{s=t}= \mu_t $$ and $$\frac{d}{ds}
Var(X_s|\Sigma_t)|_{s=t}= \sigma_t\sigma_t^T$$ I thought this is trivial
and attempted to prove it myself, however it took me nearly an hour, and
can't even show the first one..
I tried the following way, $$\frac{d}{ds} E(X_s|\Sigma_t)|_{s=t}=
\frac{d}{ds}
E(X_0+\int_0^s\mu_sds+\int_0^s\sigma_sdW_s|\Sigma_t)|_{s=t}$$$$=\frac{d}{ds}
E(X_0|\Sigma_t)+\frac{d}{ds}E(\int_0^s\mu_sds)+\frac{d}{ds}E(\int_0^s\sigma_sdW_s|\Sigma_t))$$
Then I feel that the last term should be zero by the independent increment
property, but not 100% sure I can use here, inside the integral there is
$\sigma_s$, then I don't know how to proceed from here and get the above
two equation.
Any help will be extremely appreciated!!