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ln((x^ two)/(one -x^ two))
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Similar expressions
ln((x^2)/(1+x^2))

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

Derivative of ln((x^2)/(1-x^2))

The solution

You have entered [src]

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

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

log(x^2/(1 - x^2))

Detail solution

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

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

$- \frac{2}{x^{3} - x}$

The answer is:

$- \frac{2}{x^{3} - x}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

  /         2          4                                         \
  |      5*x        4*x         /         2  \     /         2  \|
  |1 - ------- + ----------     |        x   |     |        x   ||
  |          2            2   2*|-1 + -------|   2*|-1 + -------||
  |    -1 + x    /      2\      |           2|     |           2||
  |              \-1 + x /      \     -1 + x /     \     -1 + x /|
2*|------------------------ - ---------------- + ----------------|
  |            2                        2                2       |
  \           x                   -1 + x                x        /

$$2 \left(- \frac{2 \left(\frac{x^{2}}{x^{2} - 1} - 1\right)}{x^{2} - 1} + \frac{2 \left(\frac{x^{2}}{x^{2} - 1} - 1\right)}{x^{2}} + \frac{\frac{4 x^{4}}{\left(x^{2} - 1\right)^{2}} - \frac{5 x^{2}}{x^{2} - 1} + 1}{x^{2}}\right)$$

Simplify

The third derivative [src]

  /    /         2          4   \                        /         2          4   \     /         2          4   \                   \
  |    |      3*x        2*x    |     /         2  \     |      5*x        4*x    |     |      5*x        4*x    |     /         2  \|
  |  6*|1 - ------- + ----------|     |        x   |   2*|1 - ------- + ----------|   2*|1 - ------- + ----------|     |        x   ||
  |    |          2            2|   3*|-1 + -------|     |          2            2|     |          2            2|   3*|-1 + -------||
  |    |    -1 + x    /      2\ |     |           2|     |    -1 + x    /      2\ |     |    -1 + x    /      2\ |     |           2||
  |    \              \-1 + x / /     \     -1 + x /     \              \-1 + x / /     \              \-1 + x / /     \     -1 + x /|
4*|- ---------------------------- - ---------------- - ---------------------------- + ---------------------------- + ----------------|
  |                  2                      2                        2                                2                        2     |
  \            -1 + x                      x                        x                           -1 + x                   -1 + x      /
--------------------------------------------------------------------------------------------------------------------------------------
                                                                  x

$$\frac{4 \left(\frac{3 \left(\frac{x^{2}}{x^{2} - 1} - 1\right)}{x^{2} - 1} - \frac{6 \left(\frac{2 x^{4}}{\left(x^{2} - 1\right)^{2}} - \frac{3 x^{2}}{x^{2} - 1} + 1\right)}{x^{2} - 1} + \frac{2 \left(\frac{4 x^{4}}{\left(x^{2} - 1\right)^{2}} - \frac{5 x^{2}}{x^{2} - 1} + 1\right)}{x^{2} - 1} - \frac{3 \left(\frac{x^{2}}{x^{2} - 1} - 1\right)}{x^{2}} - \frac{2 \left(\frac{4 x^{4}}{\left(x^{2} - 1\right)^{2}} - \frac{5 x^{2}}{x^{2} - 1} + 1\right)}{x^{2}}\right)}{x}$$

Simplify

The graph

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Identical expressions

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Derivative of ln((x^2)/(1-x^2))

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Piecewise:

The solution