Derivative of (x-2)(ln(x-1)-ln(x+1))

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(x - 2)*(log(x - 1) - log(x + 1))

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

(x - 2)*(log(x - 1) - log(x + 1))

Detail solution

Apply the product rule:

$\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$

$f{\left(x \right)} = x - 2$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $x - 2$ term by term:
  1. Apply the power rule: $x$ goes to $1$
  2. The derivative of the constant $-2$ is zero.
  The result is: $1$
$g{\left(x \right)} = \log{\left(x - 1 \right)} - \log{\left(x + 1 \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $\log{\left(x - 1 \right)} - \log{\left(x + 1 \right)}$ term by term:
  1. Let $u = x - 1$ .
  2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x - 1\right)$ :
    1. Differentiate $x - 1$ term by term:
      1. Apply the power rule: $x$ goes to $1$
      2. The derivative of the constant $-1$ is zero.
      The result is: $1$
    The result of the chain rule is:
    
    $\frac{1}{x - 1}$
  4. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Let $u = x + 1$ .
    2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
    3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x + 1\right)$ :
      1. Differentiate $x + 1$ term by term:
        
        Apply the power rule: $x$ goes to $1$
        
        The derivative of the constant $1$ is zero.
        
        The result is: $1$
      The result of the chain rule is:
      
      $\frac{1}{x + 1}$
    So, the result is: $- \frac{1}{x + 1}$
  The result is: $- \frac{1}{x + 1} + \frac{1}{x - 1}$
The result is: $\left(x - 2\right) \left(- \frac{1}{x + 1} + \frac{1}{x - 1}\right) + \log{\left(x - 1 \right)} - \log{\left(x + 1 \right)}$
Now simplify:

$\frac{2 x + \left(x - 1\right) \left(x + 1\right) \left(\log{\left(x - 1 \right)} - \log{\left(x + 1 \right)}\right) - 4}{\left(x - 1\right) \left(x + 1\right)}$

The answer is:

$\frac{2 x + \left(x - 1\right) \left(x + 1\right) \left(\log{\left(x - 1 \right)} - \log{\left(x + 1 \right)}\right) - 4}{\left(x - 1\right) \left(x + 1\right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

      3          3                  /   1           1    \
- --------- + -------- - 2*(-2 + x)*|-------- - ---------|
          2          2              |       3           3|
  (-1 + x)    (1 + x)               \(1 + x)    (-1 + x) /

$$- 2 \left(x - 2\right) \left(\frac{1}{\left(x + 1\right)^{3}} - \frac{1}{\left(x - 1\right)^{3}}\right) + \frac{3}{\left(x + 1\right)^{2}} - \frac{3}{\left(x - 1\right)^{2}}$$

Simplify

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Derivative of (x-2)(ln(x-1)-ln(x+1))

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The solution