Derivation of the Black-Scholes PDE

A stock price follows a geometric Brownian motion:

dS_t = \mu S_t \, dt + \sigma S_t \, dW_t \quad (\mu, \sigma > 0)

Let $V(t, S)$ be the price of a derivative written on $S$ .

Question: Derive the partial differential equation satisfied by $V$ .

Answer