# D: Understanding the output of 'n' th derivative of a function involving Exp

I am trying to find the $$n^{th$$ derivate of a function involving Exp 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 Binomial[n, K] \!$$\*SubscriptBox[\(\[PartialD]$$, $${ \*SubscriptBox[\(n$$, $$1$$], n - K}\)]
\*FractionBox[$$1$$, $$b + drift\[Gamma] + a\ \*SuperscriptBox[\(E$$, $$\(-t$$\
\*SubscriptBox[$$n$$, $$1$$]\)]\)]\), {K, 0, n}]


Posting the picture of the output since the question is more about understanding the output than the code: It looks like a sum of n terms with Binomial coefficients. But what does the partial derivative symbol represent?

• 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}] – Bob Hanlon Oct 15 at 2:56
• @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. – gaganso Oct 15 at 3:02

Whenver output is confusing, a good place to start is to look at the InputForm or FullForm:
% // InputForm

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