Solution
= Solution
{c}
Let $a=\sigma\sqrt\tau$, so $d_+=d_-+a$. Squaring gives
$$
d_+^2-d_-^2=2ad_-+a^2=2\log(S/K)+2r\tau.
$$
Exponentiation proves
$$
\boxed{Se^{-d_+^2/2}=Ke^{-r\tau}e^{-d_-^2/2}.}
$$
At fixed $S,t,T$, only $\log(S/K)$ depends on $K$, so the <partial derivatives> are
$$
\boxed{\frac{\partial d_+}{\partial K}=\frac{\partial d_-}{\partial K}
=-\frac1{K\sigma\sqrt\tau}.}
$$