Another logarithmic diff. problem and graphing question

[crm08]

Homework Statement

Find a formula for f^(n)(x) if f(x) = ln(x-1)

Homework Equations

The Attempt at a Solution

Not completely sure what the problem is asking for, any suggestions? Also, another quick question. If I am trying to graph a problem using a ti-89 that shows two tangent lines to the curve y = (ln(x))/x at the points (1,0) and (e,1/e), I found the slope function by taking the derivative, I found the equations of the two lines which are y = x-1 and y = 1/e but I don't know how to tell my calculator to graph "1/e" without it trying to raise e to a power, there is no button that is simply "e". I tried using y = 1/(1+x)^(1/x) but it gave me a bunch of random lines

 

[dirk_mec1]

To prevent confusion: do you mean nth derative or f(x) to the power n?

 

[crm08]

The way it looks to me is that it is trying to ask for a function f(x) to a power, the "n" is shown as a superscript inside parenthesis between the f and (x), I've never seen a problem written this way and there are no other problems that resemble it in the book so I'm guessing that's what it's asking

 

[crm08]

just realized it's an odd problem so the answer is in the back of the book:

f^(n)(x) = ((-1)^(n-1)*(n-1)!) / ((x-1)^n)

...I have no clue how to show the work to get there though

 

[dirk_mec1]

Calculate f'(x) then f"(x), f'''(x) until you see a pattern.

 

[crm08]

sorry it's been a few semesters since summations, would you mind writing that part in words to help me out, isn't it like the sum of something as the k goes from 0 to infinity, sorry it's gettin pretty late my mind isn't working to good right now

 

[crm08]

The pattern I see is that the exponent in the denominator is increasing by one each time although the expression stays the same, and if you multiply the numerator with the degree of the denominator, it gives the numerator for the next derivative

 

[crm08]

also the exclamation point in the answer is throwing me of, my understanding is that it means to multiply a number, for instance 5, like 5 * 4 * 3 * 2 * 1, can you explain what it means in this problem

 

[crm08]

never mind I went back to an old book, I think I got it now, thanks for the help