Someone recently asked me how to calculate the partial derivative of the Black–Scholes option price with respect to the strike. Indeed, that kind of calculation easily leads to a complete mess. You end up with page after page of hopelessly complicated expressions. You need to simplify, but how?
Fortunately, there is a trick. Express the option price as a function of d1, and write d2 as d1 minus sigma times the square root of remaining time to maturity. It turns out that the derivative of the option price with respect to d1 is zero. So when you calculate the partial derivative, for example with respect to the strike, by the chain rule, the effects that come from the dependence of d1 on the strike will disappear.
Sure enough, this trick is explained on pages 216–217 of my 1999 book, Pricing and Hedging of Derivative Securities
Low-brow, but useful.