Calculus route to digital option prices

Sketch how calculus yields Black-Scholes prices for an asset-or-nothing digital (pays $S_T$ if $S_T > K$) and a cash-or-nothing digital (pays $1$ if $S_T > K$).

Cash-or-nothing $= e^{-rT}N(d_2)$; asset-or-nothing $= S_0 N(d_1)$. Write each as a risk-neutral expectation integrated over the lognormal density above $K$: the cash digital is just the discounted probability $\mathbb{Q}(S_T > K)$, and for the asset digital completing the square in the exponent (equivalently, switching to the stock measure) gives $S_0N(d_1)$. Alternatively, differentiate the call price: $-\partial C/\partial K = e^{-rT}N(d_2)$, and the call is asset digital minus $K$ cash digitals.