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

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

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

log((x + 1)/(x - 1))

Detail solution

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

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

$- \frac{2}{x^{2} - 1}$

The answer is:

$- \frac{2}{x^{2} - 1}$

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The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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

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