Perhaps you need the algebra of random variables. By using this algebras and the standard techniques of calculus you can, at least in principle, determine the PDF of functions $f(X_1, \dots, X_n)$ of $n$ random variables from their PDF: this of course includes the standard difference and sum of random variables but also the product and quotient. A now classical text is [1] which, moreover, is available at the Internet Archive (with some restrictions).
Reference
[1] Melvin Dale Springer, The algebra of random variables (English) Wiley Series in Probability and mathematical Statistics. New York etc.: John Wiley & Sons, pp. XIX+470 (1979), ISBN:0-471-01406-0, MR519342, Zbl 0399.60002.