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((x^2-1)/(x^2+1))^4
Derivada de la función/ (x^2-1)/(x^2+1)/ x^2-1/ x^2+1/ ((x^2-1)/(x^2+1))^4

Derivada de ((x^2-1)/(x^2+1))^4

Función f() - derivada -er orden en el punto
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Gráfico:

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Solución

Ha introducido [src] 
        4
/ 2    \ 
|x  - 1| 
|------| 
| 2    | 
\x  + 1/
(x2−1x2+1)4\left(\frac{x^{2} - 1}{x^{2} + 1}\right)^{4} 
((x^2 - 1)/(x^2 + 1))^4
Solución detallada

  1. Sustituimos u=x2−1x2+1u = \frac{x^{2} - 1}{x^{2} + 1}.

  2. Según el principio, aplicamos: u4u^{4} tenemos 4u34 u^{3}

  3. Luego se aplica una cadena de reglas. Multiplicamos por ddxx2−1x2+1\frac{d}{d x} \frac{x^{2} - 1}{x^{2} + 1}:

    1. Se aplica la regla de la derivada parcial:

      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)=x2−1f{\left(x \right)} = x^{2} - 1 y g(x)=x2+1g{\left(x \right)} = x^{2} + 1.

      Para calcular ddxf(x)\frac{d}{d x} f{\left(x \right)}:

      1. diferenciamos x2−1x^{2} - 1 miembro por miembro:

        1. La derivada de una constante −1-1 es igual a cero.

        2. Según el principio, aplicamos: x2x^{2} tenemos 2x2 x

        Como resultado de: 2x2 x

      Para calcular ddxg(x)\frac{d}{d x} g{\left(x \right)}:

      1. diferenciamos x2+1x^{2} + 1 miembro por miembro:

        1. La derivada de una constante 11 es igual a cero.

        2. Según el principio, aplicamos: x2x^{2} tenemos 2x2 x

        Como resultado de: 2x2 x

      Ahora aplicamos la regla de la derivada de una divesión:

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

    Como resultado de la secuencia de reglas:

    4(x2−1)3(−2x(x2−1)+2x(x2+1))(x2+1)2(x2+1)3\frac{4 \left(x^{2} - 1\right)^{3} \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)^{3}}

  4. Simplificamos:

    16x(x2−1)3(x2+1)5\frac{16 x \left(x^{2} - 1\right)^{3}}{\left(x^{2} + 1\right)^{5}}

Respuesta:

16x(x2−1)3(x2+1)5\frac{16 x \left(x^{2} - 1\right)^{3}}{\left(x^{2} + 1\right)^{5}}

Gráfica
02468-8-6-4-2-10105-5
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Primera derivada [src] 
        4                                 
/ 2    \           /             / 2    \\
\x  - 1/  / 2    \ | 8*x     8*x*\x  - 1/|
---------*\x  + 1/*|------ - ------------|
        4          | 2                2  |
/ 2    \           |x  + 1    / 2    \   |
\x  + 1/           \          \x  + 1/   /
------------------------------------------
                   2                      
                  x  - 1
(x2−1)4(x2+1)4(x2+1)(−8x(x2−1)(x2+1)2+8xx2+1)x2−1\frac{\frac{\left(x^{2} - 1\right)^{4}}{\left(x^{2} + 1\right)^{4}} \left(x^{2} + 1\right) \left(- \frac{8 x \left(x^{2} - 1\right)}{\left(x^{2} + 1\right)^{2}} + \frac{8 x}{x^{2} + 1}\right)}{x^{2} - 1}
Simplificar
Segunda derivada [src] 
             /                                                                                                                /           2\\
             |                                                                                                    2 /      2\ |     -1 + x ||
             |                                                                                             2   2*x *\-1 + x /*|-1 + -------||
           2 |          /          2       2       2 /      2\\        /           2\        /           2\                   |           2||
  /      2\  |/      2\ |    -1 + x     4*x     4*x *\-1 + x /|      2 |     -1 + x |      2 |     -1 + x |                   \      1 + x /|
8*\-1 + x / *|\-1 + x /*|1 - ------- - ------ + --------------| + 2*x *|-1 + -------| + 8*x *|-1 + -------|  - -----------------------------|
             |          |          2        2             2   |        |           2|        |           2|                     2           |
             |          |     1 + x    1 + x      /     2\    |        \      1 + x /        \      1 + x /                1 + x            |
             \          \                         \1 + x /    /                                                                             /
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                                                                          4                                                                  
                                                                  /     2\                                                                   
                                                                  \1 + x /
8(x2−1)2(−2x2(x2−1)(x2−1x2+1−1)x2+1+8x2(x2−1x2+1−1)2+2x2(x2−1x2+1−1)+(x2−1)(4x2(x2−1)(x2+1)2−4x2x2+1−x2−1x2+1+1))(x2+1)4\frac{8 \left(x^{2} - 1\right)^{2} \left(- \frac{2 x^{2} \left(x^{2} - 1\right) \left(\frac{x^{2} - 1}{x^{2} + 1} - 1\right)}{x^{2} + 1} + 8 x^{2} \left(\frac{x^{2} - 1}{x^{2} + 1} - 1\right)^{2} + 2 x^{2} \left(\frac{x^{2} - 1}{x^{2} + 1} - 1\right) + \left(x^{2} - 1\right) \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)\right)}{\left(x^{2} + 1\right)^{4}}
Simplificar
Tercera derivada [src] 
               /                                                                                                                                                                                                                                                      2 /          2       2       2 /      2\\                                                                                   2 /          2       2       2 /      2\\                                                       2                                           \
               |                                                                                                                                                                                                                        2 /           2\     /      2\  |    -1 + x     2*x     2*x *\-1 + x /|                                                                          /      2\  |    -1 + x     4*x     4*x *\-1 + x /|                   /           2\        /           2\                            2 /           2\|
               |                                                                                                                                                                                                               /      2\  |     -1 + x |   6*\-1 + x / *|1 - ------- - ------ + --------------|                                                                        2*\-1 + x / *|1 - ------- - ------ + --------------|       2 /      2\ |     -1 + x |      2 |     -1 + x |  /      2\      2 /      2\  |     -1 + x ||
               |                                              2                    /                     2                                    2\                                                                               \-1 + x / *|-1 + -------|                |          2        2             2   |                                                                                     |          2        2             2   |   12*x *\-1 + x /*|-1 + -------|   8*x *|-1 + -------| *\-1 + x /   8*x *\-1 + x / *|-1 + -------||
               |          /           2\        /           2\      /           2\ |            /      2\        2 /      2\       2 /      2\ |               /          2       2       2 /      2\\        /           2\              |           2|                |     1 + x    1 + x      /     2\    |               /           2\ /          2       2       2 /      2\\                |     1 + x    1 + x      /     2\    |                   |           2|        |           2|                              |           2||
     /      2\ |/      2\ |     -1 + x |      2 |     -1 + x |      |     -1 + x | |        2   \-1 + x /    16*x *\-1 + x /   10*x *\-1 + x / |     /      2\ |    -1 + x     4*x     4*x *\-1 + x /|      2 |     -1 + x |              \      1 + x /                \                         \1 + x /    /     /      2\ |     -1 + x | |    -1 + x     4*x     4*x *\-1 + x /|                \                         \1 + x /    /                   \      1 + x /        \      1 + x /                              \      1 + x /|
16*x*\-1 + x /*|\-1 + x /*|-1 + -------| - 8*x *|-1 + -------|  - 4*|-1 + -------|*|-1 + 7*x  - ---------- - --------------- + ----------------| + 2*\-1 + x /*|1 - ------- - ------ + --------------| + 4*x *|-1 + -------| - ------------------------- - ---------------------------------------------------- - 4*\-1 + x /*|-1 + -------|*|1 - ------- - ------ + --------------| - ---------------------------------------------------- - ------------------------------ + ------------------------------ + ------------------------------|
               |          |           2|        |           2|      |           2| |                   2               2                  2    |               |          2        2             2   |        |           2|                  2                                        2                                      |           2| |          2        2             2   |                               2                                           2                                2                                 2           |
               |          \      1 + x /        \      1 + x /      \      1 + x / |              1 + x           1 + x           /     2\     |               |     1 + x    1 + x      /     2\    |        \      1 + x /             1 + x                                    1 + x                                       \      1 + x / |     1 + x    1 + x      /     2\    |                          1 + x                                       1 + x                            1 + x                          /     2\            |
               \                                                                   \                                              \1 + x /     /               \                         \1 + x /    /                                                                                                                                       \                         \1 + x /    /                                                                                                                                      \1 + x /            /
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                                                                                                                                                                                                                                                                           4                                                                                                                                                                                                                                                                   
                                                                                                                                                                                                                                                                   /     2\                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                   \1 + x /
16x(x2−1)(8x2(x2−1)2(x2−1x2+1−1)(x2+1)2+8x2(x2−1)(x2−1x2+1−1)2x2+1−12x2(x2−1)(x2−1x2+1−1)x2+1−8x2(x2−1x2+1−1)2+4x2(x2−1x2+1−1)−(x2−1)2(x2−1x2+1−1)x2+1−6(x2−1)2(2x2(x2−1)(x2+1)2−2x2x2+1−x2−1x2+1+1)x2+1−2(x2−1)2(4x2(x2−1)(x2+1)2−4x2x2+1−x2−1x2+1+1)x2+1−4(x2−1)(x2−1x2+1−1)(4x2(x2−1)(x2+1)2−4x2x2+1−x2−1x2+1+1)+(x2−1)(x2−1x2+1−1)+2(x2−1)(4x2(x2−1)(x2+1)2−4x2x2+1−x2−1x2+1+1)−4(x2−1x2+1−1)(10x2(x2−1)2(x2+1)2−16x2(x2−1)x2+1+7x2−(x2−1)2x2+1−1))(x2+1)4\frac{16 x \left(x^{2} - 1\right) \left(\frac{8 x^{2} \left(x^{2} - 1\right)^{2} \left(\frac{x^{2} - 1}{x^{2} + 1} - 1\right)}{\left(x^{2} + 1\right)^{2}} + \frac{8 x^{2} \left(x^{2} - 1\right) \left(\frac{x^{2} - 1}{x^{2} + 1} - 1\right)^{2}}{x^{2} + 1} - \frac{12 x^{2} \left(x^{2} - 1\right) \left(\frac{x^{2} - 1}{x^{2} + 1} - 1\right)}{x^{2} + 1} - 8 x^{2} \left(\frac{x^{2} - 1}{x^{2} + 1} - 1\right)^{2} + 4 x^{2} \left(\frac{x^{2} - 1}{x^{2} + 1} - 1\right) - \frac{\left(x^{2} - 1\right)^{2} \left(\frac{x^{2} - 1}{x^{2} + 1} - 1\right)}{x^{2} + 1} - \frac{6 \left(x^{2} - 1\right)^{2} \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 \left(x^{2} - 1\right)^{2} \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} - 4 \left(x^{2} - 1\right) \left(\frac{x^{2} - 1}{x^{2} + 1} - 1\right) \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) + \left(x^{2} - 1\right) \left(\frac{x^{2} - 1}{x^{2} + 1} - 1\right) + 2 \left(x^{2} - 1\right) \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) - 4 \left(\frac{x^{2} - 1}{x^{2} + 1} - 1\right) \left(\frac{10 x^{2} \left(x^{2} - 1\right)^{2}}{\left(x^{2} + 1\right)^{2}} - \frac{16 x^{2} \left(x^{2} - 1\right)}{x^{2} + 1} + 7 x^{2} - \frac{\left(x^{2} - 1\right)^{2}}{x^{2} + 1} - 1\right)\right)}{\left(x^{2} + 1\right)^{4}}
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Gráfico
Derivada de ((x^2-1)/(x^2+1))^4
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