Probability integral transform states that if a random variable $X$ has a continuous distribution for which the cumulative distribution function (CDF) is $F_X$, then $F_X(X)$ has a standard uniform distribution, that is, $F_X(X)\sim U(0,1).$
My question is about its pdf instead of cdf.
Question: Given a random variable $X$ and its density function $f(x),$ what is the distribution of $f(X)$?
I have a feeling that $f(X)$ does not have a uniform distribution as density is the derivative of CDF. But I do not know what is the derivative of a uniform distribution.