$F'_X(g(x))=f_X(x)g'(x)$

Question

I am doing exercises through a book and I see on some explanations on exercises that $$F_X'(g(x))=f_X(g(x))g'(x)$$

Where $F_X(x)$ is the distribution function and $f_X(x)$ is the probability density function. I did not see any proof of this, and can't seem to find it on here.

So my question is: Is this always true? How can I prove this?

Answer 1

What you have written is wrong. $F_X'(g(x))=f_X(g(x))$. However $(F_X(g(x))' =f_X(g(x)) g'(x)$ by Chain Rule.