Undergraduate-level Black-Scholes derivation

Give a derivation of the Black-Scholes equation that a student who knows basic calculus and probability, but no stochastic analysis, could follow.

Hedge away the randomness and demand the riskless rate. Model the stock as $dS=\mu S\,dt+\sigma S\,dW$ and treat $(dW)^2$ as $dt$, so a Taylor expansion of $V(S,t)$ gives $dV=(V_t+\mu SV_S+\tfrac12\sigma^2S^2V_{SS})dt+\sigma SV_S\,dW$. A portfolio long the option and short $V_S$ shares has no $dW$ term, so to avoid arbitrage it must grow at $r$, which yields $V_t+\tfrac12\sigma^2S^2V_{SS}+rSV_S-rV=0$.