Differentiate a Variable Limit Ito Integral Consider a variable limit integral $I(t)=\int\limits_0^{\phi(t)}M(s)dW(s)$, 
where $\phi(t)$ is an increasing deterministic function with $\phi(0)=0$, the integrand $M(t)$ is stochastic, and $W(t)$ is a standard Brownian motion. Assume that $M(t)$ and $W(t)$ are adapted to filtration $\mathscr{F}_t$.
I am unsure whether I can differentiate this integral as usual, i.e. $dI(t)=M(\phi(t))\phi^{'}(t)dW(t)$.
If not, what should I do to get the limit of the integral rid of those functions.

I think you cannot write it in this form, but using random time change, you can write it in another form as follows. By Theorem 8.5.7 of Oksendall's book, you can write
$$I_t = \int_0^t M_{\phi(s)}\sqrt{\phi'(s)}d\tilde B_s,$$
where $\tilde B_t = \int_0^{\phi(t)} \sqrt{(\phi^{-1})'(s)}dW_s$ is another Brownian motion (let $\alpha_t:=\phi(t)$ in the theorem). In this form, you can write 
$$dI_t = M_{\phi(t)}\sqrt{\phi'(t)}d\tilde B_t.$$
You don't even need that $I_t$ is adapted to the filtration of $W_t$ (which holds if and only if $\phi(t)\leq t$).
