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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:enter image description here 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 enter image description here So now, if I want to calculate the fair value of the option,enter image description here 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?enter image description here

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You need to use the leibniz integral rule. More details here: https://en.wikipedia.org/wiki/Leibniz_integral_rule

But in a nuthshell (ignoring the $e^{-r\tau}$ which is a constant):

$$\frac{\partial C}{\partial K} = -(K-K)f(x)\frac{\partial K}{\partial K} + \int^\infty_K \frac{\partial}{\partial K}(x-K)f(x)dx = -\int^\infty_K f(x)dx$$

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