Derivation of Black-Scholes Equation: Why $d(\Delta S) = \Delta dS$ When deriving the Black-Scholes equation, a key step is to construct a portfolio
\begin{equation}
\Pi = V - \Delta S
\end{equation}
that contains a long position of the option and a short position of the stock. We then take derivative on both sides and get
\begin{equation}
d\Pi = dV - \Delta dS.
\end{equation}
By doing so we implicitly assume that $\Delta$ is a constant so that $d(\Delta S) = \Delta dS$.
However later on we choose $\Delta = \frac{\partial V}{\partial S}$ to eliminate the uncertainty. Since $V$ is a function of $S$ and $t$, so is $\Delta$. In that case, shouldn't we use
\begin{equation}
d(\Delta S) = \Delta dS + Sd\Delta
\end{equation}
and then apply Ito's lemma to $d\Delta$?

In the standard derivation of the Black-Scholes model this portfolio is assumed to be self-financing that is there are no inflows or outflows of money. The formal definition of a self-financing portfolio in your case is that
$$d\Pi=dV-\Delta dS.$$
That is the self-financing assumption explains why you do not differentiate w.r.t. $\Delta$. For an intuitive explanation of the self-financing condition, see e.g. here Definition of self-financing strategy.
