Stating and deriving the BSM equation

State the Black-Scholes-Merton partial differential equation for a derivative on a non-dividend stock and outline how it is derived.

$\tfrac{\partial f}{\partial t}+rS\tfrac{\partial f}{\partial S}+\tfrac12\sigma^2S^2\tfrac{\partial^2f}{\partial S^2}=rf$. With $dS=\mu S\,dt+\sigma S\,dW$, Itô's lemma gives $df$; a portfolio short one derivative and long $\partial f/\partial S$ shares has no Brownian term, so it must earn the risk-free rate over each instant, and equating its change to $r$ times its value yields the PDE.