# How to derive the Breeden-Litzenberger formula?

I was going through a proof of the Breeden-Litzenberger formula and I was stuck on one of the intermediate steps.

The pdf for the prices of the underlying at expiry is defined as $$f(x)$$ and $$S_T$$ is the price of the underlying at expiry and $$K$$ is the strike price so the probability that the underlying expires at a price higher than the strike is: Moreover, the fair value of the option is the expected payout of the option at expiry, discounted by the risk-free rate till expiry, or mathematically as So now, if I want to calculate the fair value of the option, Now, if I differentiate the call option value with respect to the strike, how do I arrive at this answer shown below and how does $$(x-K)$$ just disappear?

But in a nuthshell (ignoring the $$e^{-r\tau}$$ which is a constant):
$$\frac{\partial C}{\partial K} = -(K-K)f(x)\frac{\partial K}{\partial K} + \int^\infty_K \frac{\partial}{\partial K}(x-K)f(x)dx = -\int^\infty_K f(x)dx$$