The usual proof of the product rule is:
\begin{align*} (fg)'(x) &= \lim_{h \rightarrow 0}\frac{f(x+h)g(x+h) - f(x)g(x)}{h} \\&= \lim_{h \rightarrow 0}\frac{f(x+h)g(x+h) - f(x+h)g(x) + f(x+h)g(x) - f(x)g(x)}{h} \\ &= \lim_{h \rightarrow 0}\left(\frac{f(x+h)(g(x+h)-g(x))}{h} + \frac{g(x)(f(x+h)-f(x))}{h}\right) \tag 1 \\ &= \lim_{h \rightarrow 0}f(x+h)\frac{g(x+h)-g(x)}{h} + \lim_{h \rightarrow 0}g(x)\frac{f(x+h)-f(x)}{h} \tag 2 \\ &= f(x)g'(x) + g(x)f'(x).\tag 3 \end{align*}
Why isn't that proof is usually written backwards?
Transition from (1) to (2) can only be done with the knowledge that corresponding parts of (1) exist (same for (2) to (3)).
Also we are supposed to show that (fg)' exist and equals to (3), given that f' and g' exist. So isn't it more clear to start with (3)?
To me proof above looks more like a "working backwards" sketch.
