Deriving the BS PDE

First we start from the geometric brownian motion that the stock is assumed to follow.
\[dS=\mu Sdt+\sigma SdW
\]
or
\[
\frac{dS}{S}=\mu dt+\sigma dW
\]
with $\mu$ the drift of the stock, $\sigma$ the volatility. $dW$ represent the random process driving the stock, it is a Wiener process.
Options are derivatives instruments as they are a functions of the stock price and time.
From Itô's lemma we know that given a process
\[
dx=a(x,t)dt+b(x,t)dW
\]
and a function $G(x,t)$, we have
\[
dG=(\frac{\partial G}{\partial x}a+\frac{\partial G}{\partial t}+\frac{1}{2}\frac{\partial^2 G}{\partial x^2}b^2)dt+\frac{\partial G}{\partial x}bdW
\]
Applied to an option $V$, function of our stock $S$ and time $t$, we get
\[
dV=(\frac{\partial V}{\partial S}\mu S+\frac{\partial V}{\partial t}+\frac{1}{2}\frac{\partial^2 V}{\partial S^2}\sigma^2S^2)dt+\frac{\partial V}{\partial S}\sigma SdW
\]
\[
dV=\frac{\partial V}{\partial S}(\mu S+\sigma SdW)+\frac{\partial V}{\partial t}dt+\frac{1}{2}\frac{\partial^2 V}{\partial S^2}\sigma^2S^2dt
\]
\[
dV=\frac{\partial V}{\partial S}dS+\frac{\partial V}{\partial t}dt+\frac{1}{2}\frac{\partial^2 V}{\partial S^2}\sigma^2S^2dt
\]
We see that the price of our derivative is affected by the randomness $dW$ through the $dS$ term.
Now say we build a portfolio made of 1 derivative and $-\partial V/\partial S$ shares of $S$.
Let's call the price of this portfolio $\Pi$. So we have
\[
\Pi=V-\frac{\partial V}{\partial S}S
\]
and
\[
d\Pi=dV-\frac{\partial V}{\partial S}dS
\]
\[
d\Pi=\frac{\partial V}{\partial t}dt+\frac{1}{2}\frac{\partial^2 V}{\partial S^2}\sigma^2S^2dt
\]
The $dW$ term is gone, the portfolio has no randomness anymore. So it's evolution with respect to time should be equal to placing this amount on a risk free account (where it would grow with the continuous rate $r$)
\[
d\Pi=r\Pi dt=\frac{\partial V}{\partial t}dt+\frac{1}{2}\frac{\partial^2 V}{\partial S^2}\sigma^2S^2dt
\]
Plugging back the value of $\Pi$ and rearranging gives
\[
\frac{\partial V}{\partial t}+\frac{1}{2}\frac{\partial^2 V}{\partial S^2}\sigma^2S^2+rS\frac{\partial V}{\partial S}-rV=0
\]
That is the generic equation that derivatives must follow in the Black and Scholes model. Note that the drift $\mu$ disappeared.
This is also the equation we discretized for our PDE pricing.