Call delta when the stock pays a continuous dividend

A stock pays a continuous dividend yield $q$ and otherwise follows Black-Scholes dynamics. Derive $\partial C/\partial S$ for a European call on it with strike $K$ and time to expiry $\tau$.

$\Delta=e^{-q\tau}N(d_1)$, where $d_1=\frac{\ln(S/K)+(r-q+\sigma^2/2)\tau}{\sigma\sqrt\tau}$. The price is $C=Se^{-q\tau}N(d_1)-Ke^{-r\tau}N(d_2)$. Differentiating, the terms from $\partial d_1/\partial S$ and $\partial d_2/\partial S$ cancel because $Se^{-q\tau}n(d_1)=Ke^{-r\tau}n(d_2)$, leaving only $e^{-q\tau}N(d_1)$.