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(x(x^2+1)/(x^2-1)^2)^(1/3)

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Derivada de la función/ x^2-1/ (x^2+1)/(x^2-1)/ x^2+1/ (x(x^2+1)/(x^2-1)^2)^(1/3)

Derivada de (x(x^2+1)/(x^2-1)^2)^(1/3)

Función f() - derivada -er orden en el punto
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
       ____________
      /   / 2    \ 
     /  x*\x  + 1/ 
    /   ---------- 
   /            2  
3 /     / 2    \   
\/      \x  - 1/
x(x2+1)(x2−1)23\sqrt[3]{\frac{x \left(x^{2} + 1\right)}{\left(x^{2} - 1\right)^{2}}} 
((x*(x^2 + 1))/(x^2 - 1)^2)^(1/3)
Solución detallada

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

  2. Según el principio, aplicamos: u3\sqrt[3]{u} tenemos 13u23\frac{1}{3 u^{\frac{2}{3}}}

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

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

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

      1. Se aplica la regla de la derivada de una multiplicación:

        ddxf(x)g(x)=f(x)ddxg(x)+g(x)ddxf(x)\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(x)=xf{\left(x \right)} = x; calculamos ddxf(x)\frac{d}{d x} f{\left(x \right)}:

        1. Según el principio, aplicamos: xx tenemos 11

        g(x)=x2+1g{\left(x \right)} = x^{2} + 1; calculamos 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

        Como resultado de: 3x2+13 x^{2} + 1

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

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

      2. Según el principio, aplicamos: u2u^{2} tenemos 2u2 u

      3. Luego se aplica una cadena de reglas. Multiplicamos por ddx(x2−1)\frac{d}{d x} \left(x^{2} - 1\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

        Como resultado de la secuencia de reglas:

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

      Ahora aplicamos la regla de la derivada de una divesión:

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

    Como resultado de la secuencia de reglas:

    −2x2(x2+1)(2x2−2)+(x2−1)2(3x2+1)3(x(x2−1)2)23(x2−1)4(x2+1)23\frac{- 2 x^{2} \left(x^{2} + 1\right) \left(2 x^{2} - 2\right) + \left(x^{2} - 1\right)^{2} \left(3 x^{2} + 1\right)}{3 \left(\frac{x}{\left(x^{2} - 1\right)^{2}}\right)^{\frac{2}{3}} \left(x^{2} - 1\right)^{4} \left(x^{2} + 1\right)^{\frac{2}{3}}}

  4. Simplificamos:

    −4x2(x2+1)+(x2−1)(3x2+1)3(x(x2−1)2)23(x2−1)3(x2+1)23\frac{- 4 x^{2} \left(x^{2} + 1\right) + \left(x^{2} - 1\right) \left(3 x^{2} + 1\right)}{3 \left(\frac{x}{\left(x^{2} - 1\right)^{2}}\right)^{\frac{2}{3}} \left(x^{2} - 1\right)^{3} \left(x^{2} + 1\right)^{\frac{2}{3}}}

Respuesta:

−4x2(x2+1)+(x2−1)(3x2+1)3(x(x2−1)2)23(x2−1)3(x2+1)23\frac{- 4 x^{2} \left(x^{2} + 1\right) + \left(x^{2} - 1\right) \left(3 x^{2} + 1\right)}{3 \left(\frac{x}{\left(x^{2} - 1\right)^{2}}\right)^{\frac{2}{3}} \left(x^{2} - 1\right)^{3} \left(x^{2} + 1\right)^{\frac{2}{3}}}

Gráfica
02468-8-6-4-2-1010-5050
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Primera derivada [src] 
       ____________                                        
      /   / 2    \          2 /         2       2 / 2    \\
     /  x*\x  + 1/  / 2    \  |  1 + 3*x     4*x *\x  + 1/|
    /   ---------- *\x  - 1/ *|----------- - -------------|
   /            2             |          2              3 |
3 /     / 2    \              |  / 2    \       / 2    \  |
\/      \x  - 1/              \3*\x  - 1/     3*\x  - 1/  /
-----------------------------------------------------------
                           / 2    \                        
                         x*\x  + 1/
x(x2+1)(x2−1)23(x2−1)2(−4x2(x2+1)3(x2−1)3+3x2+13(x2−1)2)x(x2+1)\frac{\sqrt[3]{\frac{x \left(x^{2} + 1\right)}{\left(x^{2} - 1\right)^{2}}} \left(x^{2} - 1\right)^{2} \left(- \frac{4 x^{2} \left(x^{2} + 1\right)}{3 \left(x^{2} - 1\right)^{3}} + \frac{3 x^{2} + 1}{3 \left(x^{2} - 1\right)^{2}}\right)}{x \left(x^{2} + 1\right)}
Simplificar
Segunda derivada [src] 
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                   |                                                 |        2   4*x *\1 + x /|           2   4*x *\1 + x /     |        2   4*x *\1 + x /|                   |       2   4*x *\1 + x /| |        2   4*x *\1 + x /||
                   |                                               4*|-1 - 3*x  + -------------|   -1 - 3*x  + -------------   2*|-1 - 3*x  + -------------|                   |1 + 3*x  - -------------|*|-1 - 3*x  + -------------||
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    /  ---------- *|2 - ----------- - ----------- - ------------ - ----------------------------- + ------------------------- + ----------------------------- + ------------- - ------------------------------------------------------|
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\/     \-1 + x /   \                                                                                                                                             \-1 + x /                                                           /
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                                                                                                                     2/3                                                                                                              
                                                                                                             /     2\                                                                                                                 
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x(x2−1)23(−8x23(x2−1)+8x2(x2+1)(x2−1)2+2+2(−3x2+4x2(x2+1)x2−1−1)3(x2+1)−8(x2+1)3(x2−1)−4(3x2+1)3(x2−1)−4(−3x2+4x2(x2+1)x2−1−1)3(x2−1)+−3x2+4x2(x2+1)x2−1−13x2−(−3x2+4x2(x2+1)x2−1−1)(3x2−4x2(x2+1)x2−1+1)9x2(x2+1))(x2+1)23\frac{\sqrt[3]{\frac{x}{\left(x^{2} - 1\right)^{2}}} \left(- \frac{8 x^{2}}{3 \left(x^{2} - 1\right)} + \frac{8 x^{2} \left(x^{2} + 1\right)}{\left(x^{2} - 1\right)^{2}} + 2 + \frac{2 \left(- 3 x^{2} + \frac{4 x^{2} \left(x^{2} + 1\right)}{x^{2} - 1} - 1\right)}{3 \left(x^{2} + 1\right)} - \frac{8 \left(x^{2} + 1\right)}{3 \left(x^{2} - 1\right)} - \frac{4 \left(3 x^{2} + 1\right)}{3 \left(x^{2} - 1\right)} - \frac{4 \left(- 3 x^{2} + \frac{4 x^{2} \left(x^{2} + 1\right)}{x^{2} - 1} - 1\right)}{3 \left(x^{2} - 1\right)} + \frac{- 3 x^{2} + \frac{4 x^{2} \left(x^{2} + 1\right)}{x^{2} - 1} - 1}{3 x^{2}} - \frac{\left(- 3 x^{2} + \frac{4 x^{2} \left(x^{2} + 1\right)}{x^{2} - 1} - 1\right) \left(3 x^{2} - \frac{4 x^{2} \left(x^{2} + 1\right)}{x^{2} - 1} + 1\right)}{9 x^{2} \left(x^{2} + 1\right)}\right)}{\left(x^{2} + 1\right)^{\frac{2}{3}}}
Simplificar
Tercera derivada [src] 
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      ____________ |     |                     2   |      |    -1 + x     -1 + x        -1 + x       /      2\      /      2\         /      2\         /      2\   |      |       -1 + x       -1 + x     -1 + x        /      2\   |         |       -1 + x       -1 + x     -1 + x        /      2\   |        |                     2   |        |                     2   |      |                     2   |     |           2| |                     2   |     \     -1 + x / |       -1 + x       -1 + x     -1 + x        /      2\   |      |                     2   |        |       -1 + x       -1 + x     -1 + x        /      2\   |     |           2| |                     2   |     \              -1 + x    / |       -1 + x       -1 + x     -1 + x        /      2\   |     \               -1 + x    / |      -1 + x      -1 + x      /      2\   |     |                    2   | |                     2   |      |                    2   | |                     2   |      |           2| |                     2   |         |                     2   |   |           2| |                    2   | |                     2   |      |                    2   | |                     2   ||
     /     x       |     \               -1 + x    /      \                                          \-1 + x /      \-1 + x /         \-1 + x /         \-1 + x /   /      \                                             \-1 + x /   /         \                                             \-1 + x /   /        \               -1 + x    /        \               -1 + x    /      \               -1 + x    /     \     -1 + x / \               -1 + x    /                    \                                             \-1 + x /   /      \               -1 + x    /        \                                             \-1 + x /   /     \     -1 + x / \               -1 + x    /                                \                                             \-1 + x /   /                                 \                              \-1 + x /   /     \              -1 + x    / \               -1 + x    /      \              -1 + x    / \               -1 + x    /      \     -1 + x / \               -1 + x    /         \               -1 + x    /   \     -1 + x / \              -1 + x    / \               -1 + x    /      \              -1 + x    / \               -1 + x    /|
    /  ---------- *|- ------------------------------ + -------------------------------------------------------------------------------------------------------------- + -------------------------------------------------------------- - ----------------------------------------------------------------- - -------------------------------- - -------------------------------- - ------------------------------ - -------------------------------------------- + ---------------------------------------------------------------------------- + ------------------------------ + ---------------------------------------------------------------- - -------------------------------------------- - ---------------------------------------------------------------------------------------- + -------------------------------------------------------------------------- + -------------------------------------------------------- + --------------------------------------------------------- + --------------------------------------------- + --------------------------------- + --------------------------------------------------------------------- - ---------------------------------------------------------|
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3 /    /      2\   |                x                                                                                                                                                                                                                                 -1 + x                                            /      2\                          /     2\                          x*\1 + x /                                  x                                                                                                                 x*\-1 + x /                                          1 + x                                                  x*\1 + x /                                                           x*\1 + x /                                                                          x*\1 + x /                                                         x *\1 + x /                                                   /     2\                                            x*\-1 + x /                            \1 + x /*\-1 + x /                                       x *\1 + x /                                                   x*\1 + x /*\-1 + x /                  |
\/     \-1 + x /   \                                                                                                                                                                                                                                                                                                    \-1 + x /                          \1 + x /                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            x*\1 + x /                                                                                                                                                                                                                                                /
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                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      27*\1 + x /
x(x2−1)23(60x(4x2x2−1−12x2(x2+1)(x2−1)2−3+4(x2+1)x2−1+2(3x2+1)x2−1)x2+1−60x(−3x2+4x2(x2+1)x2−1−1)(x2+1)2−144x(4x2x2−1−12x2(x2+1)(x2−1)2−3+4(x2+1)x2−1+2(3x2+1)x2−1)x2−1+120x(−3x2+4x2(x2+1)x2−1−1)(x2−1)(x2+1)−72x(−3x2+4x2(x2+1)x2−1−1)(x2−1)2+6(4x2x2−1−1)(4x2x2−1−12x2(x2+1)(x2−1)2−3+4(x2+1)x2−1+2(3x2+1)x2−1)x+36(4x2x2−1−12x2(x2+1)(x2−1)2−3+4(x2+1)x2−1+2(3x2+1)x2−1)x+18(48x4(x2−1)2−96x4(x2+1)(x2−1)3−44x2x2−1+60x2(x2+1)(x2−1)2+12x2(3x2+1)(x2−1)2+3−4(x2+1)x2−1−2(3x2+1)x2−1)x−6(4x2x2−1−1)(−3x2+4x2(x2+1)x2−1−1)x(x2+1)+6(−3x2+4x2(x2+1)x2−1−1)(8x2x2−1−12x2(x2+1)(x2−1)2−3+6(x2+1)x2−1)x(x2+1)−12(−3x2+4x2(x2+1)x2−1−1)x(x2+1)−6(3x2−4x2(x2+1)x2−1+1)(4x2x2−1−12x2(x2+1)(x2−1)2−3+4(x2+1)x2−1+2(3x2+1)x2−1)x(x2+1)+10(−3x2+4x2(x2+1)x2−1−1)(3x2−4x2(x2+1)x2−1+1)x(x2+1)2+12(4x2x2−1−1)(−3x2+4x2(x2+1)x2−1−1)x(x2−1)+36(−3x2+4x2(x2+1)x2−1−1)x(x2−1)−24(−3x2+4x2(x2+1)x2−1−1)(3x2−4x2(x2+1)x2−1+1)x(x2−1)(x2+1)−3(4x2x2−1−1)(−3x2+4x2(x2+1)x2−1−1)x3−18(−3x2+4x2(x2+1)x2−1−1)x3+(4x2x2−1−1)(−3x2+4x2(x2+1)x2−1−1)(3x2−4x2(x2+1)x2−1+1)x3(x2+1)+6(−3x2+4x2(x2+1)x2−1−1)(3x2−4x2(x2+1)x2−1+1)x3(x2+1))27(x2+1)23\frac{\sqrt[3]{\frac{x}{\left(x^{2} - 1\right)^{2}}} \left(\frac{60 x \left(\frac{4 x^{2}}{x^{2} - 1} - \frac{12 x^{2} \left(x^{2} + 1\right)}{\left(x^{2} - 1\right)^{2}} - 3 + \frac{4 \left(x^{2} + 1\right)}{x^{2} - 1} + \frac{2 \left(3 x^{2} + 1\right)}{x^{2} - 1}\right)}{x^{2} + 1} - \frac{60 x \left(- 3 x^{2} + \frac{4 x^{2} \left(x^{2} + 1\right)}{x^{2} - 1} - 1\right)}{\left(x^{2} + 1\right)^{2}} - \frac{144 x \left(\frac{4 x^{2}}{x^{2} - 1} - \frac{12 x^{2} \left(x^{2} + 1\right)}{\left(x^{2} - 1\right)^{2}} - 3 + \frac{4 \left(x^{2} + 1\right)}{x^{2} - 1} + \frac{2 \left(3 x^{2} + 1\right)}{x^{2} - 1}\right)}{x^{2} - 1} + \frac{120 x \left(- 3 x^{2} + \frac{4 x^{2} \left(x^{2} + 1\right)}{x^{2} - 1} - 1\right)}{\left(x^{2} - 1\right) \left(x^{2} + 1\right)} - \frac{72 x \left(- 3 x^{2} + \frac{4 x^{2} \left(x^{2} + 1\right)}{x^{2} - 1} - 1\right)}{\left(x^{2} - 1\right)^{2}} + \frac{6 \left(\frac{4 x^{2}}{x^{2} - 1} - 1\right) \left(\frac{4 x^{2}}{x^{2} - 1} - \frac{12 x^{2} \left(x^{2} + 1\right)}{\left(x^{2} - 1\right)^{2}} - 3 + \frac{4 \left(x^{2} + 1\right)}{x^{2} - 1} + \frac{2 \left(3 x^{2} + 1\right)}{x^{2} - 1}\right)}{x} + \frac{36 \left(\frac{4 x^{2}}{x^{2} - 1} - \frac{12 x^{2} \left(x^{2} + 1\right)}{\left(x^{2} - 1\right)^{2}} - 3 + \frac{4 \left(x^{2} + 1\right)}{x^{2} - 1} + \frac{2 \left(3 x^{2} + 1\right)}{x^{2} - 1}\right)}{x} + \frac{18 \left(\frac{48 x^{4}}{\left(x^{2} - 1\right)^{2}} - \frac{96 x^{4} \left(x^{2} + 1\right)}{\left(x^{2} - 1\right)^{3}} - \frac{44 x^{2}}{x^{2} - 1} + \frac{60 x^{2} \left(x^{2} + 1\right)}{\left(x^{2} - 1\right)^{2}} + \frac{12 x^{2} \left(3 x^{2} + 1\right)}{\left(x^{2} - 1\right)^{2}} + 3 - \frac{4 \left(x^{2} + 1\right)}{x^{2} - 1} - \frac{2 \left(3 x^{2} + 1\right)}{x^{2} - 1}\right)}{x} - \frac{6 \left(\frac{4 x^{2}}{x^{2} - 1} - 1\right) \left(- 3 x^{2} + \frac{4 x^{2} \left(x^{2} + 1\right)}{x^{2} - 1} - 1\right)}{x \left(x^{2} + 1\right)} + \frac{6 \left(- 3 x^{2} + \frac{4 x^{2} \left(x^{2} + 1\right)}{x^{2} - 1} - 1\right) \left(\frac{8 x^{2}}{x^{2} - 1} - \frac{12 x^{2} \left(x^{2} + 1\right)}{\left(x^{2} - 1\right)^{2}} - 3 + \frac{6 \left(x^{2} + 1\right)}{x^{2} - 1}\right)}{x \left(x^{2} + 1\right)} - \frac{12 \left(- 3 x^{2} + \frac{4 x^{2} \left(x^{2} + 1\right)}{x^{2} - 1} - 1\right)}{x \left(x^{2} + 1\right)} - \frac{6 \left(3 x^{2} - \frac{4 x^{2} \left(x^{2} + 1\right)}{x^{2} - 1} + 1\right) \left(\frac{4 x^{2}}{x^{2} - 1} - \frac{12 x^{2} \left(x^{2} + 1\right)}{\left(x^{2} - 1\right)^{2}} - 3 + \frac{4 \left(x^{2} + 1\right)}{x^{2} - 1} + \frac{2 \left(3 x^{2} + 1\right)}{x^{2} - 1}\right)}{x \left(x^{2} + 1\right)} + \frac{10 \left(- 3 x^{2} + \frac{4 x^{2} \left(x^{2} + 1\right)}{x^{2} - 1} - 1\right) \left(3 x^{2} - \frac{4 x^{2} \left(x^{2} + 1\right)}{x^{2} - 1} + 1\right)}{x \left(x^{2} + 1\right)^{2}} + \frac{12 \left(\frac{4 x^{2}}{x^{2} - 1} - 1\right) \left(- 3 x^{2} + \frac{4 x^{2} \left(x^{2} + 1\right)}{x^{2} - 1} - 1\right)}{x \left(x^{2} - 1\right)} + \frac{36 \left(- 3 x^{2} + \frac{4 x^{2} \left(x^{2} + 1\right)}{x^{2} - 1} - 1\right)}{x \left(x^{2} - 1\right)} - \frac{24 \left(- 3 x^{2} + \frac{4 x^{2} \left(x^{2} + 1\right)}{x^{2} - 1} - 1\right) \left(3 x^{2} - \frac{4 x^{2} \left(x^{2} + 1\right)}{x^{2} - 1} + 1\right)}{x \left(x^{2} - 1\right) \left(x^{2} + 1\right)} - \frac{3 \left(\frac{4 x^{2}}{x^{2} - 1} - 1\right) \left(- 3 x^{2} + \frac{4 x^{2} \left(x^{2} + 1\right)}{x^{2} - 1} - 1\right)}{x^{3}} - \frac{18 \left(- 3 x^{2} + \frac{4 x^{2} \left(x^{2} + 1\right)}{x^{2} - 1} - 1\right)}{x^{3}} + \frac{\left(\frac{4 x^{2}}{x^{2} - 1} - 1\right) \left(- 3 x^{2} + \frac{4 x^{2} \left(x^{2} + 1\right)}{x^{2} - 1} - 1\right) \left(3 x^{2} - \frac{4 x^{2} \left(x^{2} + 1\right)}{x^{2} - 1} + 1\right)}{x^{3} \left(x^{2} + 1\right)} + \frac{6 \left(- 3 x^{2} + \frac{4 x^{2} \left(x^{2} + 1\right)}{x^{2} - 1} - 1\right) \left(3 x^{2} - \frac{4 x^{2} \left(x^{2} + 1\right)}{x^{2} - 1} + 1\right)}{x^{3} \left(x^{2} + 1\right)}\right)}{27 \left(x^{2} + 1\right)^{\frac{2}{3}}}
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Gráfico
Derivada de (x(x^2+1)/(x^2-1)^2)^(1/3)
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