How can I find the formal derivative of a polynomial w.r.t. to its coefficients

Asked Aug 07 '15 at 01:22

Active Aug 07 '15 at 06:45

Viewed 93 times

I want to define a function $f(x)$ as in

$\quad \quad f(x) = \sum^{n}_{i=0} c_i x^i$

as a symbolic expression. Then I'd like to do operations on it. For example, take derivatives of it with respect to any variable or collection of variables. Is there a concise way to do this, so that I get the results in symbolic form?

Again, for example, if I would like to get $ x^i$ from evaluating $ \frac{d f(x)}{dc_i}$.

edited Aug 07 '15 at 06:45

m_goldberg

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asked Aug 07 '15 at 01:22

Charlie Parker

Is $n$ a fixed number, or do you want to leave it symbolic? If $n$ is symbolic, then have you seen this? – J. M.'s missing motivation Aug 07 '15 at 01:27
@Guesswhoitis. I wasn't really planning to take derivatives with respect n, so fixed would be ok, for a specific $f(x)$. I wish I knew more about mathematica to know how that matters/makes a difference. I am more concerned to take derivatives wrt to $c_i$ and $x^i$. – Charlie Parker Aug 07 '15 at 01:31
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Then, it's easy to just do D[Sum[C[k] x^k, {k, 0, 5}], x]. – J. M.'s missing motivation Aug 07 '15 at 01:50
@Guesswhoitis. doesn't quite work as the example I gave. If I do D[Sum[C[k] x^k, {k, 0, 5}], C[k]] it returns zeros, should return $x^k$. But its definitively a good start! :D – Charlie Parker Aug 07 '15 at 01:53
That's the "leave it symbolic" case in my previous comment; in which case, I already pointed you to an applicable thread. – J. M.'s missing motivation Aug 07 '15 at 01:53
@Guesswhoitis. I will take a look at the thread then. Thanks for the help Guess who it is...miley cyrus? :p – Charlie Parker Aug 07 '15 at 01:54
That's because there is no C[k] to differentiate against; only C[0], C[1], … and heavens, I do not jut out my posterior like that in public! – J. M.'s missing motivation Aug 07 '15 at 01:55
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@Guesswhoitis. Time to change your display name. How about: "You all know who it is"? – m_goldberg Aug 07 '15 at 06:14
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@m_goldberg so he is Miley? – shrx Aug 07 '15 at 09:03

How can I find the formal derivative of a polynomial w.r.t. to its coefficients

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