How do I write a proof that it is possible to obtain the product rule from chain rule, sum rule and from $\frac{d}{dx}x^2=2x$
Thank you
2025-01-13 00:13:49.1736727229
write proof that it is possible to obtain the product rule from chain, sum rule
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The hint $(x+y)^2-(x-y)^2$ was enough for the OP, great. Decided to make the hint into an "answer" for the sake of pointing out that it was an interesting question, at least to me.
Interesting because the same trick is a standard thing in an inner-product space, deducing things about the inner product from things about the norm. Never realized it also had application to calculus...
Details added on request: Say $f$ and $g$ are differentiable. Then $$4(fg)'=((f+g)^2-(f-g)^2)'=2(f+g)(f+g)'-2(f-g)(f-g)'=4f'g+4fg'.$$