The Black–Scholes equation, via Itô's lemma

Apply Itô's lemma to $V(S_t,t)$, treating $V$ as a smooth function of the random process $S_t$ and of time. It states $dV = \left(\dfrac{\partial V}{\partial t} + \mu S_t \dfrac{\partial V}{\partial S} + \dfrac{1}{2}\sigma^2 S_t^2 \dfrac{\partial^2 V}{\partial S^2}\right)dt + \sigma S_t \dfrac{\partial V}{\partial S}\,dW_t$. Unlike ordinary calculus, an extra $\frac{1}{2}\sigma^2 S_t^2 \partial^2 V/\partial S^2$ term appears, coming from the nonzero quadratic variation of Brownian motion.

Now build a hedged portfolio $\Pi = V - \Delta S_t$, holding one derivative and short $\Delta$ shares of stock, with $\Delta$ chosen to be $\partial V/\partial S$. Its change is $d\Pi = dV - \Delta\,dS_t$. Substituting the expression for $dV$ and $dS_t = \mu S_t\,dt + \sigma S_t\,dW_t$, and choosing $\Delta = \partial V/\partial S$, the random terms $\sigma S_t \partial V/\partial S\,dW_t$ and $-\Delta \sigma S_t\,dW_t$ cancel exactly.

What remains is $d\Pi = \left(\dfrac{\partial V}{\partial t} + \dfrac{1}{2}\sigma^2 S_t^2 \dfrac{\partial^2 V}{\partial S^2}\right)dt$, a purely deterministic (riskless) change over the instant $dt$, with no dependence left on the stock's drift $\mu$.

A riskless portfolio must earn exactly the risk-free rate, or an arbitrage would be possible by borrowing or lending against it. So $d\Pi = r\Pi\,dt = r(V - S_t\,\partial V/\partial S)\,dt$. Equating the two expressions for $d\Pi$ and dividing by $dt$ gives $\dfrac{\partial V}{\partial t} + \dfrac{1}{2}\sigma^2 S^2 \dfrac{\partial^2 V}{\partial S^2} = rV - rS\dfrac{\partial V}{\partial S}$, which rearranges into the Black–Scholes equation.