Ito's Lemma gives that for a function $F$ of a stochastic variable $X$, $dF = \frac{dF}{dX}dX + \frac{1}{2}\frac{d^2F}{dX^2}dt$
Given a stochastic differential equation $dS = a(S) dt + b(S) dX$ and a function $V(S)$, the differential equation satisfied by $V(S)$ is stated to be $dV = \frac{dV}{dS}dS + \frac{1}{2}b^2 \frac{d^2V}{dS^2}dt $. The book says this can be derived properly or the cheating substitution of $dX^2 = dt$ could be used.
But I just don't understand how the expression for $dV$ is obtained - either properly or by using the Taylor series. While using the Taylor series, how should I expand $V$? Should I use the Taylor series for a composite function, as shown, for instance, here? And how might I do it "properly"?
The book is Quantitative Finance of Paul Wilmott, I have made a screenshot of the relevant section.
I make a completely naive attempt by trying to apply Ito's lemma to $V$ treating $S$ as the random variable (since it is, in effect the function of one).
$dV = \frac{dV}{dS}dS + \frac{1}{2}\frac{d^2V}{dS^2}dS^2 $
and then substituting the given expression of $dS$ in the second term of the RHS. $dV = \frac{dV}{dS}dS + \frac{1}{2}\frac{d^2V}{dS^2}(a(S) dt + b(S) dX)^2$
This can directly give the correct expression for $dV$ if the $dt$ term inside the braces can be made to go to zero. But why would that be? So obviously there's something subtle going on that I am completely missing. Any pointers would be appreciated. Also regarding how to derive it properly.