just out of curiosity, how is adam.chlipala.net/cpdt a tutorial? It seems to be an entire book! How did you specifically had in mind to go through it?

actually Ian, I don't understand your question. Are you taking the derivative of P with respect to x (the sample) or $\theta$? I can't conceive how it makes sense to take the derivative wrt to x if x is a sample. If its a sample, then its observed and fixed, so taking the derivative wrt to it is simply zero. Or am I missing something?

now that Im re-reading the question I think I misunderstood it. It seems that the issue was that he couldn't take the derivative of of x, of course if x is an observed sample its a fixed number so taking a derivative of it leads to 0. It seems that taking the derivative of $\theta$ remains sensible even with samples observed (i.e. its similar to MLE, maximum likelihood estimation).

I read your answer but I didn't understand it. Why is my counter example wrong. Consider $P_{\theta}(X = x) = \theta^x (1 - \theta)^{1-x}$ and let $x=1$ be the sample observed. Then $P_{\theta}(X = 1) = \theta $. One can easily take the derivative of that function, it has $\theta$ as a variable, its derivative is simply 1. Why is that not correct?

in the end, what did you end up telling them to convince them?

@NieldeBeaudrap sorry if this is a simple question, but what do u mean by "proof virus"? Not sure if I understood ur comment.

@usul, I tried asking in now, I am not sure if its good enough now, what do you think? cstheory.stackexchange.com/questions/30591/…

@BjørnKjos-Hanssen thank you, finally a comment that is very useful. I will look into this.

@usul I don't know why it was put on hold either, as the people that voted, left no comments and no signals for me to improve my question.

@JoonasIlmavirta I didn't know what good tags to add. But it seems people already added them.