Inactive derivative: extract constants

Question

I have a problem with an unevaluated derivative. In the example code below, i need the integration wrt to y be evaluated, while the differentation wrt x needs to stay unevaluated. I see the point that it is exactly the purpose of Inactive[ ] to hold the function to be differentiated "as is", but i would like to know if there is a workaround to "extract" anything not dependent on x from the inactive derivative. Any help would be highly appreciated.

Integrate[Cos[y]*Inactive[D][-Cos[y]*f[x]*Inactive[D][f[x], x], x], {y, 0, 2*Pi}]

Would be duplicates, mutatis mutandis: replacement rule to pull independent expression outside of Integrate, Pull Constants outside of integrals. — Michael E2, Sep 02 '14 at 12:27

Teake Nutma · Accepted Answer · 2014-09-02T12:09:28.267

2

Assuming expr is the integral in the question, we can manually extract any x independent terms from the differentiation as follows:

expr //. RuleDelayed[
  Inactive[D][Times[indep_ , rest___], x_] /; FreeQ[indep, x],
  indep * Inactive[D][Times[rest], x]
]

- π Inactive[D][f[x] Inactive[D][f[x], x], x]

On second thought, perhaps Dt is better suited for your purposes?

expr //. Inactive[D][arg_, x] :> Dt[arg, x, Constants -> {y}]

- π ( f'[x]^2 + f[x] f''[x] )

edited Sep 02 '14 at 12:09

answered Sep 02 '14 at 11:48

Teake Nutma

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Thank you, that was exactly what I was looking for. – Alex Sep 02 '14 at 11:58
I think the OP might be trying to prevent the product rule from being applied (but wants the scalar multiple rule to be applied). If so, your second thought would not produce the desired output. +1 for the first thought, though. :) – Michael E2 Sep 02 '14 at 12:26

Inactive derivative: extract constants

1 Answers1