Past exam of the mathematical models of financial derivatives course of the University of Oxford / 2015 / 3 / a / i / Solution

Let $a=\sigma \sqrt{\tau }$, so $d_{+}=d_{-}+a$. Squaring givesExponentiation provesAt fixed $S,t,T$, only $\log (S/K)$ depends on $K$, so the partial derivatives are