Proofs of the multiplication or chain rule for derivatives that invoke symmetry

Question

Introductory calculus texts sometimes include direct proofs of the multiplication and chain rules for derivatives by:

1. Introducing a pair of differences $D_f=\frac{f(x+h)-f(x)}{h}-f'(x)$ and $D_g=\frac{g(y+k)-g(y)}{k}-g'(y)$
2. Expressing $f(x+h)$ and $g(y+k)$ in terms of $D_f$ and $D_g$
3. Substituting those expressions in the definition of derivative for the rule
4. Noting that $D_f$ goes to $0$ with $h$ and $D_g$ goes to $0$ with $k$ and $h$ goes to $0$ with $k$.

I don't know if it's just me but these proofs seem awfully unsatisfying.

More insightful is the motivation for the substitutions (say, in terms of linear approximations at a point).

But I wonder now if there is a route to a different kind of proof that takes advantage of the "interchangability-symmetry" of $f$ and $g$ in the case of $f g$ or even in the case of composition. For products, it really seems like there could be some symmetry-based argument when you consider the formula $f'g + g'f$.

So, my question is, is there an alternative proof or even just another way to look at this in terms of symmetry?