Another logarithmic diff. problem and graphing question

• crm08
In summary, the conversation is about finding the formula for the nth derivative of f(x) = ln(x-1). The solution involves taking successive derivatives and observing a pattern in the exponents in the denominator. The final answer is ((-1)^(n-1)*(n-1)!) / ((x-1)^n).
crm08

Homework Statement

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

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

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

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

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

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

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

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

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

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

1. What is a logarithmic differential problem?

A logarithmic differential problem is a type of mathematical problem that involves solving for the variable in a logarithmic equation. This often requires using the properties of logarithms and differentiating the equation to find the solution.

2. How do you solve a logarithmic differential problem?

To solve a logarithmic differential problem, you must first rewrite the equation in its logarithmic form and then use the properties of logarithms to simplify it. Next, you can take the derivative of both sides of the equation and solve for the variable. Finally, you can check your solution by plugging it back into the original equation.

3. What is the purpose of graphing a logarithmic differential equation?

Graphing a logarithmic differential equation allows you to visualize the relationship between the variables and see how the equation changes over a range of values. This can help you identify the behavior of the function and find any critical points or asymptotes.

4. How do you graph a logarithmic differential equation?

To graph a logarithmic differential equation, you can use a graphing calculator or plot points manually by choosing values for the variable and plugging them into the equation. You can also use the properties of logarithms to transform the equation into a linear form and then graph the resulting line.

5. Are there any real-life applications of logarithmic differential equations?

Yes, logarithmic differential equations have various real-life applications, such as modeling population growth, radioactive decay, and chemical reactions. They are also commonly used in finance and economics to study compound interest and growth rates.

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