Derivative of ln(x/(x-1))

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   /  x  \
log|-----|
   \x - 1/

$$\log{\left(\frac{x}{x - 1} \right)}$$

log(x/(x - 1))

Detail solution

Let $u = \frac{x}{x - 1}$ .
The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{x}{x - 1}$ :
1. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = x$ and $g{\left(x \right)} = x - 1$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Apply the power rule: $x$ goes to $1$
  To find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Differentiate $x - 1$ term by term:
    1. The derivative of the constant $-1$ is zero.
    2. Apply the power rule: $x$ goes to $1$
    The result is: $1$
  Now plug in to the quotient rule:
  
  $- \frac{1}{\left(x - 1\right)^{2}}$
The result of the chain rule is:

$- \frac{x - 1}{x \left(x - 1\right)^{2}}$
Now simplify:

$- \frac{1}{x \left(x - 1\right)}$

The answer is:

$- \frac{1}{x \left(x - 1\right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

        /  1        x    \
(x - 1)*|----- - --------|
        |x - 1          2|
        \        (x - 1) /
--------------------------
            x

$$\frac{\left(x - 1\right) \left(- \frac{x}{\left(x - 1\right)^{2}} + \frac{1}{x - 1}\right)}{x}$$

Simplify

The second derivative [src]

/       x   \ /1     1   \
|-1 + ------|*|- + ------|
\     -1 + x/ \x   -1 + x/
--------------------------
            x

$$\frac{\left(\frac{x}{x - 1} - 1\right) \left(\frac{1}{x - 1} + \frac{1}{x}\right)}{x}$$

Simplify

The third derivative [src]

  /       x   \ /  1        1           1     \
2*|-1 + ------|*|- -- - --------- - ----------|
  \     -1 + x/ |   2           2   x*(-1 + x)|
                \  x    (-1 + x)              /
-----------------------------------------------
                       x

$$\frac{2 \left(\frac{x}{x - 1} - 1\right) \left(- \frac{1}{\left(x - 1\right)^{2}} - \frac{1}{x \left(x - 1\right)} - \frac{1}{x^{2}}\right)}{x}$$

Simplify

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Derivative of ln(x/(x-1))

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The solution