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Deriving The Product Rule from Symmetry and Units
(f⋅g)' ≠ f'⋅g' (inconsistent units)
Simplest expression with correct units: f'g or g'f
The LHS has a "swapping" symmetry f ⇔ g, hence so must the RHS.
Therefore we must 𝒂𝒅𝒅 thempic.twitter.com/3HtCOtVMHA
Nice observation by @jasnyder610.
On the last line, the most general form of the RHS is actually (f⋅g)' = 𝑐⋅(f'g + g'f), for some constant 𝑐.
We can solve for 𝑐 by setting g(x) = 1, and we then see that we must have that 𝑐=1.
This is really neat. When I see things like this, it makes me a little sad that by plunging myself deep down the mathematical rabbit hole, I seem to have let my basic physical intuition atrophy to the point that I forget to connect things back to reality.
Thanks! it's how I think of things as a physicist.. we need all the cross checks we can get given how sloppy we are ;)
Neat! I like it. There's also the matter of an overall scale factor, which you can see has to be 1 by (for example) setting g(x) = 1
Ah good point!
Do you have any other intuitive explanations of the product rule? Particularly in the complex case. This visualization explains very neatly the real case, but I've yet to see a good visual for the complex case.pic.twitter.com/ynOO1PyWd3
this is cool! did you make it?
I posed the problem of finding functions f,g satisfying the wrong product rule. (Also in relation to convolution, for what it's worth!)
Nice, but it still doesn't help me memorizing the *quotient* rule.
Yes but it's interesting to note that perhaps the lack of symmetry in the quotient rule is why it's more complicated
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