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Similar expressions
1/4*ln((x^2+1)/(x^2+1))
1/4*ln((x^2-1)/(x^2-1))

The derivative of the function/ 1/4*ln((x^2-1)/(x^2+1))

Derivative of 1/4*ln((x^2-1)/(x^2+1))

The solution

You have entered [src]

   / 2    \
   |x  - 1|
log|------|
   | 2    |
   \x  + 1/
-----------
     4

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

  /   / 2    \\
  |   |x  - 1||
  |log|------||
  |   | 2    ||
d |   \x  + 1/|
--|-----------|
dx\     4     /

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

Transform

Detail solution

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

$\frac{x}{x^{4} - 1}$

The answer is:

$\frac{x}{x^{4} - 1}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

         /             / 2    \\
/ 2    \ | 2*x     2*x*\x  - 1/|
\x  + 1/*|------ - ------------|
         | 2                2  |
         |x  + 1    / 2    \   |
         \          \x  + 1/   /
--------------------------------
             / 2    \           
           4*\x  - 1/

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

Simplify

The second derivative [src]

                            /           2\        /           2\                 
                          2 |     -1 + x |      2 |     -1 + x |                 
                       2*x *|-1 + -------|   2*x *|-1 + -------|                 
          2       2         |           2|        |           2|      2 /      2\
    -1 + x     4*x          \      1 + x /        \      1 + x /   4*x *\-1 + x /
1 - ------- - ------ - ------------------- + ------------------- + --------------
          2        2               2                     2                   2   
     1 + x    1 + x           1 + x                -1 + x            /     2\    
                                                                     \1 + x /    
---------------------------------------------------------------------------------
                                     /      2\                                   
                                   2*\-1 + x /

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

Simplify

The third derivative [src]

   /                                /          2       2       2 /      2\\     /          2       2       2 /      2\\     /          2       2       2 /      2\\                                            \ 
   |           2              2     |    -1 + x     4*x     4*x *\-1 + x /|     |    -1 + x     4*x     4*x *\-1 + x /|     |    -1 + x     2*x     2*x *\-1 + x /|        /           2\        /           2\| 
   |     -1 + x         -1 + x    2*|1 - ------- - ------ + --------------|   2*|1 - ------- - ------ + --------------|   6*|1 - ------- - ------ + --------------|      2 |     -1 + x |      2 |     -1 + x || 
   |-1 + -------   -1 + -------     |          2        2             2   |     |          2        2             2   |     |          2        2             2   |   4*x *|-1 + -------|   4*x *|-1 + -------|| 
   |           2              2     |     1 + x    1 + x      /     2\    |     |     1 + x    1 + x      /     2\    |     |     1 + x    1 + x      /     2\    |        |           2|        |           2|| 
   |      1 + x          1 + x      \                         \1 + x /    /     \                         \1 + x /    /     \                         \1 + x /    /        \      1 + x /        \      1 + x /| 
-x*|------------ - ------------ - ----------------------------------------- + ----------------------------------------- + ----------------------------------------- + ------------------- - -------------------| 
   |        2              2                             2                                           2                                           2                                  2        /     2\ /      2\| 
   |   1 + x         -1 + x                         1 + x                                      -1 + x                                       1 + x                          /      2\         \1 + x /*\-1 + x /| 
   \                                                                                                                                                                       \-1 + x /                           / 
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                           2                                                                                                     
                                                                                                     -1 + x

$$- \frac{x \left(- \frac{4 x^{2} \left(\frac{x^{2} - 1}{x^{2} + 1} - 1\right)}{\left(x^{2} - 1\right) \left(x^{2} + 1\right)} + \frac{4 x^{2} \left(\frac{x^{2} - 1}{x^{2} + 1} - 1\right)}{\left(x^{2} - 1\right)^{2}} + \frac{\frac{x^{2} - 1}{x^{2} + 1} - 1}{x^{2} + 1} + \frac{6 \cdot \left(\frac{2 x^{2} \left(x^{2} - 1\right)}{\left(x^{2} + 1\right)^{2}} - \frac{2 x^{2}}{x^{2} + 1} - \frac{x^{2} - 1}{x^{2} + 1} + 1\right)}{x^{2} + 1} - \frac{2 \cdot \left(\frac{4 x^{2} \left(x^{2} - 1\right)}{\left(x^{2} + 1\right)^{2}} - \frac{4 x^{2}}{x^{2} + 1} - \frac{x^{2} - 1}{x^{2} + 1} + 1\right)}{x^{2} + 1} - \frac{\frac{x^{2} - 1}{x^{2} + 1} - 1}{x^{2} - 1} + \frac{2 \cdot \left(\frac{4 x^{2} \left(x^{2} - 1\right)}{\left(x^{2} + 1\right)^{2}} - \frac{4 x^{2}}{x^{2} + 1} - \frac{x^{2} - 1}{x^{2} + 1} + 1\right)}{x^{2} - 1}\right)}{x^{2} - 1}$$

Simplify

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Derivative of 1/4*ln((x^2-1)/(x^2+1))

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Piecewise:

The solution

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Derivative of 1​/4*ln((x^2-1)/(x^2+1))​

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Piecewise:

The solution

Derivative of 1/4*ln((x^2-1)/(x^2+1))