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I am trying to find the $n^{th$ derivate of a function involving Exp[1] as given below.

w = (a* Exp[-t Subscript[n, 1]] + 
     b)/((a* Exp[-t Subscript[n, 1]] + b) + drift\[Gamma]);
D[w, {Subscript[n, 1], n}]

I get the following output after running the above code:

Inactive[Sum][a E^(-t Subscript[n, 1]) (-t)^K[1] Binomial[n, K[1]] \!\(
\*SubscriptBox[\(\[PartialD]\), \({
\*SubscriptBox[\(n\), \(1\)], n - K[1]}\)]
\*FractionBox[\(1\), \(b + drift\[Gamma] + a\ 
\*SuperscriptBox[\(E\), \(\(-t\)\ 
\*SubscriptBox[\(n\), \(1\)]\)]\)]\), {K[1], 0, n}]

Posting the picture of the output since the question is more about understanding the output than the code:

output

It looks like a sum of n terms with Binomial coefficients. But what does the partial derivative symbol represent?

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  • $\begingroup$ What version are you using? With v12 I do not get a summation. However, you appear to be overloading the symbol n. You use it both in Subscript[n, 1] and in the number of times (n+1) to take the partial derivative. Do you see a change if you use D[w, {Subscript[n, 1], m + 1}] $\endgroup$ – Bob Hanlon Oct 15 at 2:56
  • $\begingroup$ @BobHanlon, Thank you for the response. I am using v12 and the just checked to make sure the context is local to the notebook. Also, I corrected a typo D[w, {Subscript[n, 1], n}]. I get the same output if I use m. $\endgroup$ – gaganso Oct 15 at 3:02
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Whenver output is confusing, a good place to start is to look at the InputForm or FullForm:

% // InputForm

(* Inactive[Sum][(a*(-t)^K[1]*Binomial[n, K[1]]*
  D[(b + drift\[Gamma] + a/E^(t*Subscript[n, 1]))^(-1), {Subscript[n, 1], n - K[1]}])/
  E^(t*Subscript[n, 1]), {K[1], 0, n}]*)

So, the subscript on the \[PartialD] corresponds to the 2nd argument of D, exactly, and you're taking the n-K[1]'th derivative of the expression. You're essentially getting the generalized product rule, which is a sum with binomial coefficients.

BTW, the information about the meaning of the subscript, and related useful information, is in the Details and Options section of ref/D, near the top.

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