How to Find the Derivative of f(x)=ln[x/(x-1)]?

In summary, the formula for finding the derivative of a fraction is (numerator derivative * denominator) - (numerator * denominator derivative) / (denominator)^2. To find the derivative of a fraction with a variable in both the numerator and denominator, the quotient rule must be used. The derivative can be simplified if the numerator and denominator have common factors. If the denominator is a constant, the power rule can be used to find the derivative. The derivative of a complex fraction can also be found using the quotient rule.
  • #1
lreichardt
4
0
How do I take the derivative of

f(x)=ln[x/(x-1)]?

Thanks!
 
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  • #2
Welcome to PF!

Hi lreichardt! Welcome to PF! :smile:

Just use the chain rule: derivative of ln, times the derivative of x/(x-1). :smile:
 
  • #3
so, (x/1)(quotient rule of x/(x-1)? =

(x/1)(-x/x-1)=-x^2/x-1
 
  • #4
ooh dear …

better go the easy way … simplify it first …

ln(x/(x-1)) = ln(x) - ln(x-1) … now differentiate! :smile:
 
  • #5
Chain rule = The difference of the derivative of the "Outer" and derivative of the "Inner", put simply.
 
  • #6
Thanks all, figured it out.
 

1. What is the formula for finding the derivative of a fraction?

The formula for finding the derivative of a fraction is (numerator derivative * denominator) - (numerator * denominator derivative) / (denominator)^2.

2. How do you find the derivative of a fraction with a variable in both the numerator and denominator?

To find the derivative of a fraction with a variable in both the numerator and denominator, you will need to use the quotient rule. This rule states that the derivative of a fraction is equal to (numerator derivative * denominator) - (numerator * denominator derivative) / (denominator)^2.

3. Can you simplify the derivative of a fraction?

Yes, in some cases, the derivative of a fraction can be simplified. For example, if the numerator and denominator have common factors, they can be canceled out, resulting in a simpler form of the derivative.

4. How do you find the derivative of a fraction with a constant in the denominator?

If the denominator of a fraction is a constant, the derivative can be found by using the power rule. This means that the derivative of the fraction is equal to (numerator * exponent of the denominator) / (denominator)^(exponent of the denominator + 1).

5. Can you find the derivative of a complex fraction?

Yes, the derivative of a complex fraction can be found by using the quotient rule. This rule states that the derivative of a fraction is equal to (numerator derivative * denominator) - (numerator * denominator derivative) / (denominator)^2.

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