I'm an undergraduate student studying for the actuarial exams and was wondering if someone could please walk me through the proof and intuition of this please? I haven't taken an analysis course yet, I'm a senior and have taken calculus 1 through 3, linear algebra, differential equations, probability, theory of interest, and this year taking statistical inference, mathematics of finance, and number theory. I wanted to put my math background here so that someone can help me knowing the level of mathematics I have. I understand the FTC but for some reason this stumped me and can't seem to find a reference for proofing this. I linked the pictures. Thank you to anyone who has the time to help me with this, your time is much appreciated. The Y and X here are random variables but I don't think that matters since random variables are functions anyways
Figured it out it was proved in my probability book with a great proof!.
The substitution rule may be of use to you: https://en.wikipedia.org/wiki/Integration_by_substitution
In this case, (looking at the proof on Wikipedia) let $\phi$ be $g^{-1}$ $-$ that should transform it into a more FTC-friendly form but, as you may have noticed, this can only work if you are allowed to assume $g(-\infty)$ exists (so you can rewrite the lower bound as $g^{-1}(g(-\infty))$).
As one of the commenters mentioned, knowing general differentiation under the integral sign may be more broadly useful, but this should work well enough if you're not looking for anything too fancy.