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Does anyone have any nice ways of handling differentials like \[ \frac{d^2}{dx^2}\left(\frac{20}{1+x^2}\right)\]? And, the more extreme cases like \[\frac{d^4}{dx^4}\left(\frac{20}{1+x^2}\right)\].

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Have you considered using a negative exponent? \[\frac{d^2}{dx^2}(20)(1+x^2)^{-1}\]
Yes, but the problem gets ugly, quickly, especially considering we are taking the second derivative. So, I'm guessing, there is none? All just straight up computation. Hmm.
i usually use a "u-substitution" in derivatives...i suppose that makes me weird....

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Enlighten me...?
@LolWolf \[\frac{ d ^{2} }{ dx ^{2} }(\frac{ 20 }{ 1+x ^{2} })\]implies that you find the second derivative of the function y, if\[y =\frac{ 20 }{ 1+x ^{2} }\]I assume you know your derivative rules, correct?
Yes, I'm not wondering so much how to find it, but, rather, how to do it neatly.
(If someone has some ingenious way of managing it, et al)
y'=-40x/(1+x^2)^2 now do it again the second time
No, I am a math teacher and I will tell you that you must use the quotient rule to find the second derivative, but it would be wise to use the chain rule (power of a function rule) to find the first derivative.
use the quotient rule
@mark_o. Yes, I know. I'm not talking about how to find it, finding it is simple, I'm referring to taking this twice or thrice without having an utter mess of calculations, or without it taking more than some short time. I don't know if there even *is* such a way, I'm just guessing.
@mark_o. please stop giving out answers. You're not helping!
I already have the answer. That's not what I'm looking for. I'm wondering if there is some way to manage this in a quick and nice manner, rather than crunching numbers.
(Again, there might *not* be, but I'm not the most imaginative person in Calculus--I deal with Number Theory--and this question came up)
Those derivative rules are in place for a reason. You're looking for a short cut and I'm telling you that those derivative rules are the shortcut.
im sorry i didnt want to give the answer on the second derivative., i thought he didnt knoe how to arrive at the first derivative.... continue plzz
@mark_o. That's alright! He said he already got the answer so no harm done.
Yes, I am not referring to the derivative rules. Of course, most people could come up with the proofs, and I am not looking for a shortcut, at least, not directly. The point that I ask is *not* how to find them, nor to get a lesson on the shortcuts, and whether they should be used or not, I'm asking if there *is* a nice way of *handling* the operations, or if there is some much nicer (not necessarily *easier*) way of dealing with problems that are similar.
ok continue plzz...
@mark_o. there is nothing to continue LOL because @LolWolf the answer to your question is a very nice NO.
That's all I wanted to know. Thank you.
well, there no way of getting around it,or to short cut them, just continue solving them wether you started on quotient rule or product will arrive on the same answer,......
Yes, thank you.
ok YW, have fun now
Actually, there IS, indeed a much nicer way to do this derivative, if we split it into separate functions. Not to beat a dead horse, or anything, but please don't say that one 'cannot' do something without actually knowing so, even if you're a teacher. If we simply assume the function of \(x^2=l\), then, by repeated application of the chain rule, we reduce the laborious task of handling the repeated quotient rule to a simple product rule with a single multiplication per step, along with a much easier-to-compute chain rule. Enjoy.

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