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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))

Function f() - derivative -N order at the point
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The solution

You have entered [src] 
   /   2  \
   |  x   |
log|------|
   |     2|
   \1 - x /
log⁡(x21−x2)\log{\left(\frac{x^{2}}{1 - x^{2}} \right)} 
log(x^2/(1 - x^2))
Detail solution

  1. Let u=x21−x2u = \frac{x^{2}}{1 - x^{2}}.

  2. The derivative of log⁡(u)\log{\left(u \right)} is 1u\frac{1}{u}.

  3. Then, apply the chain rule. Multiply by ddxx21−x2\frac{d}{d x} \frac{x^{2}}{1 - x^{2}}:

    1. Apply the quotient rule, which is:

      ddxf(x)g(x)=−f(x)ddxg(x)+g(x)ddxf(x)g2(x)\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(x)=x2f{\left(x \right)} = x^{2} and g(x)=1−x2g{\left(x \right)} = 1 - x^{2}.

      To find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

      1. Apply the power rule: x2x^{2} goes to 2x2 x

      To find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

      1. Differentiate 1−x21 - x^{2} term by term:

        1. The derivative of the constant 11 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: x2x^{2} goes to 2x2 x

          So, the result is: −2x- 2 x

        The result is: −2x- 2 x

      Now plug in to the quotient rule:

      2x3+2x(1−x2)(1−x2)2\frac{2 x^{3} + 2 x \left(1 - x^{2}\right)}{\left(1 - x^{2}\right)^{2}}

    The result of the chain rule is:

    2x3+2x(1−x2)x2(1−x2)\frac{2 x^{3} + 2 x \left(1 - x^{2}\right)}{x^{2} \left(1 - x^{2}\right)}

  4. Now simplify:

    −2x3−x- \frac{2}{x^{3} - x}

The answer is:

−2x3−x- \frac{2}{x^{3} - x}

The graph
02468-8-6-4-2-1010-5050
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
(1−x2)(2x3(1−x2)2+2x1−x2)x2\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(−2(x2x2−1−1)x2−1+2(x2x2−1−1)x2+4x4(x2−1)2−5x2x2−1+1x2)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      /
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                                                                  x
4(3(x2x2−1−1)x2−1−6(2x4(x2−1)2−3x2x2−1+1)x2−1+2(4x4(x2−1)2−5x2x2−1+1)x2−1−3(x2x2−1−1)x2−2(4x4(x2−1)2−5x2x2−1+1)x2)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
Derivative of ln((x^2)/(1-x^2))
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