Can the chain rule and product rule be derived using Lagrange's algebraic approach to calculus?

Question

So a while ago I learned about an algebraic approach to calculus with polynomials. Basically if $P(x)=a_n(x-r)^n+a_{n-1}(x-r)^{n-1}+...+a_1(x-r)+a_0$ it can be shown, algebraically, that $$a_n=\frac{P^{(n)}(r)}{n!}$$ Of course, the concept of the nth derivative isn't actually introduced, $P^{(n)}$ represents the nth application of the power rule to $P$. So I was wondering if this theory can be extended to rational functions: $f(x)=\frac{P(x)}{Q(x)}$ where $P$ and $Q$ are polynomials. Obviously, $$f'(x)=\frac{P'(x)}{Q(x)}-\frac{P(x)Q'(x)}{Q(x)^2}$$ But I was wondering if its possible to derive this with an extension of the same algebraic method-it seems to me that it would be necessary to prove the chain and product rule, but of course I'd be glad to hear if there's another way.