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?
\[\frac{d^2}{dx^2}(20)(1+x^2)^{-1}\]

i usually use a "u-substitution" in derivatives...i suppose that makes me weird....

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