# Differentiating sum of log

• Apr 26th 2013, 01:51 AM
Katmarn
Differentiating sum of log
Hi everyone,

This is my first post, so please bear with me if I am posting in the wrong forum.

I study economics and I am reading an article. I do not understand how the author gets the following conclusions:

He takes log of this function: Attachment 28164

And ends up with: Attachment 28165

I thought it would be Attachment 28163 instead of the log[T!/(T-t)].

Secondly I want do differentiate logc_t^t, but I am unsure how to deal with the log sum expressions. How do you differentiate Attachment 28163??

Wolfram alpha uses gamma functions which I honestly do not understand. The article I am reading finds the following derivative:

Attachment 28166

I hope someone can clearify this for me. Thank you very much!

Jacob
• Apr 26th 2013, 08:49 PM
chiro
Re: Differentiating sum of log
Hey Katmarn.

Try expressing that term in terms of factorials and see if it is equivalent.
• Apr 29th 2013, 03:22 AM
Katmarn
Re: Differentiating sum of log
Thank you Chiro.

I had tried it before without luck, but as I did it again it suddently made sense :)

However, I still do not know how you take the derivative of a factorial term. Could you please show me how you do it on: Attachment 28190

When I express it in terms of factorials I get: log((1+b^(1/n)*T)!/[1+b^(1/n)*(T-t-1)]!), and I have no idea how to take the derivative of that...

Thank you!
• Apr 29th 2013, 05:17 AM
chiro
Re: Differentiating sum of log
If you are differentiating a factorial, take a look at the Euler Gamma function and use the fundamental theorem of calculus:

Gamma function - Wikipedia, the free encyclopedia
• Apr 30th 2013, 04:46 AM
Katmarn
Re: Differentiating sum of log
Thank you for the answer, but I must admit I am still lost as I do not fully understand the gamma function.

Could you, or someone else, please explain the steps the author must use to get his result?

He takes the derivative of: Attachment 28201 wrt t.

And he ends up with: Attachment 28202

I have tried to express the sum with factorials, and got: Attachment 28203

I do not know if this is even possible as 0<beta<1. However, if it is correct i can split the log up and just focus on the bottom part when i want to take the derivative wrt. t.

And here I am stuck...

All help is appreciated.
• Apr 30th 2013, 05:22 AM
chiro
Re: Differentiating sum of log
The Gamma function is the factorial function, but is extended for all real values instead of only being valid for integer values. It is not defined at 0 or the negative integers, but it is defined everywhere else.

The connection is that n! = Gamma(n+1).
• Apr 30th 2013, 05:36 AM
Katmarn
Re: Differentiating sum of log
Ok. Do you have the time to explain how the author ends up with the result that he does?
• Apr 30th 2013, 05:54 PM
chiro
Re: Differentiating sum of log
I'll take a look at this later on so if I haven't responded tomorrow, just bump the thread.
• May 2nd 2013, 01:17 AM
Katmarn
Re: Differentiating sum of log
Chiro
• May 2nd 2013, 01:57 AM
chiro
Re: Differentiating sum of log
Try substituting for s values t - 1, t - 2, ..., t - t and use the fact that d/dx(u + v) = du/dx + dv/dx.

Then collect the derivatives of all the logarithms and see if you get the same answer.