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((x*(x^2-2)^3*(x+2))/((x-1)^2*(x^3+x^2+x)))^1/5
Derivada de la función/ ((x*(x^2-2)^3*(x+2))/((x-1)^2*(x^3+x^2+x)))^1/5

Derivada de ((x*(x^2-2)^3*(x+2))/((x-1)^2*(x^3+x^2+x)))^1/5

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

Gráfico:

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Definida a trozos:

Solución

Ha introducido [src] 
       ________________________
      /            3           
     /     / 2    \            
    /    x*\x  - 2/ *(x + 2)   
   /    ---------------------- 
5 /            2 / 3    2    \ 
\/      (x - 1) *\x  + x  + x/
x(x2−2)3(x+2)(x−1)2(x+(x3+x2))5\sqrt[5]{\frac{x \left(x^{2} - 2\right)^{3} \left(x + 2\right)}{\left(x - 1\right)^{2} \left(x + \left(x^{3} + x^{2}\right)\right)}} 
(((x*(x^2 - 2)^3)*(x + 2))/(((x - 1)^2*(x^3 + x^2 + x))))^(1/5)
Solución detallada

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

  2. Según el principio, aplicamos: u5\sqrt[5]{u} tenemos 15u45\frac{1}{5 u^{\frac{4}{5}}}

  3. Luego se aplica una cadena de reglas. Multiplicamos por ddxx(x2−2)3(x+2)(x−1)2(x+(x3+x2))\frac{d}{d x} \frac{x \left(x^{2} - 2\right)^{3} \left(x + 2\right)}{\left(x - 1\right)^{2} \left(x + \left(x^{3} + x^{2}\right)\right)}:

    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(x+2)(x2−2)3f{\left(x \right)} = x \left(x + 2\right) \left(x^{2} - 2\right)^{3} y g(x)=(x−1)2(x3+x2+x)g{\left(x \right)} = \left(x - 1\right)^{2} \left(x^{3} + x^{2} + x\right).

      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)h(x)=f(x)g(x)ddxh(x)+f(x)h(x)ddxg(x)+g(x)h(x)ddxf(x)\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} h{\left(x \right)} = f{\left(x \right)} g{\left(x \right)} \frac{d}{d x} h{\left(x \right)} + f{\left(x \right)} h{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} h{\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)=(x2−2)3g{\left(x \right)} = \left(x^{2} - 2\right)^{3}; calculamos ddxg(x)\frac{d}{d x} g{\left(x \right)}:

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

        2. Según el principio, aplicamos: u3u^{3} tenemos 3u23 u^{2}

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

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

            1. La derivada de una constante −2-2 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:

          6x(x2−2)26 x \left(x^{2} - 2\right)^{2}

        h(x)=x+2h{\left(x \right)} = x + 2; calculamos ddxh(x)\frac{d}{d x} h{\left(x \right)}:

        1. diferenciamos x+2x + 2 miembro por miembro:

          1. La derivada de una constante 22 es igual a cero.

          2. Según el principio, aplicamos: xx tenemos 11

          Como resultado de: 11

        Como resultado de: 6x2(x+2)(x2−2)2+x(x2−2)3+(x+2)(x2−2)36 x^{2} \left(x + 2\right) \left(x^{2} - 2\right)^{2} + x \left(x^{2} - 2\right)^{3} + \left(x + 2\right) \left(x^{2} - 2\right)^{3}

      Para calcular ddxg(x)\frac{d}{d x} g{\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)=(x−1)2f{\left(x \right)} = \left(x - 1\right)^{2}; calculamos ddxf(x)\frac{d}{d x} f{\left(x \right)}:

        1. Sustituimos u=x−1u = x - 1.

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

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

          1. diferenciamos x−1x - 1 miembro por miembro:

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

            2. Según el principio, aplicamos: xx tenemos 11

            Como resultado de: 11

          Como resultado de la secuencia de reglas:

          2x−22 x - 2

        g(x)=x3+x2+xg{\left(x \right)} = x^{3} + x^{2} + x; calculamos ddxg(x)\frac{d}{d x} g{\left(x \right)}:

        1. diferenciamos x3+x2+xx^{3} + x^{2} + x miembro por miembro:

          1. Según el principio, aplicamos: xx tenemos 11

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

          3. Según el principio, aplicamos: x3x^{3} tenemos 3x23 x^{2}

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

        Como resultado de: (x−1)2(3x2+2x+1)+(2x−2)(x3+x2+x)\left(x - 1\right)^{2} \left(3 x^{2} + 2 x + 1\right) + \left(2 x - 2\right) \left(x^{3} + x^{2} + x\right)

      Ahora aplicamos la regla de la derivada de una divesión:

      −x(x+2)(x2−2)3((x−1)2(3x2+2x+1)+(2x−2)(x3+x2+x))+(x−1)2(x3+x2+x)(6x2(x+2)(x2−2)2+x(x2−2)3+(x+2)(x2−2)3)(x−1)4(x3+x2+x)2\frac{- x \left(x + 2\right) \left(x^{2} - 2\right)^{3} \left(\left(x - 1\right)^{2} \left(3 x^{2} + 2 x + 1\right) + \left(2 x - 2\right) \left(x^{3} + x^{2} + x\right)\right) + \left(x - 1\right)^{2} \left(x^{3} + x^{2} + x\right) \left(6 x^{2} \left(x + 2\right) \left(x^{2} - 2\right)^{2} + x \left(x^{2} - 2\right)^{3} + \left(x + 2\right) \left(x^{2} - 2\right)^{3}\right)}{\left(x - 1\right)^{4} \left(x^{3} + x^{2} + x\right)^{2}}

    Como resultado de la secuencia de reglas:

    −x(x+2)(x2−2)3((x−1)2(3x2+2x+1)+(2x−2)(x3+x2+x))+(x−1)2(x3+x2+x)(6x2(x+2)(x2−2)2+x(x2−2)3+(x+2)(x2−2)3)5(x(x+2)(x2−2)3(x−1)2(x+(x3+x2)))45(x−1)4(x3+x2+x)2\frac{- x \left(x + 2\right) \left(x^{2} - 2\right)^{3} \left(\left(x - 1\right)^{2} \left(3 x^{2} + 2 x + 1\right) + \left(2 x - 2\right) \left(x^{3} + x^{2} + x\right)\right) + \left(x - 1\right)^{2} \left(x^{3} + x^{2} + x\right) \left(6 x^{2} \left(x + 2\right) \left(x^{2} - 2\right)^{2} + x \left(x^{2} - 2\right)^{3} + \left(x + 2\right) \left(x^{2} - 2\right)^{3}\right)}{5 \left(\frac{x \left(x + 2\right) \left(x^{2} - 2\right)^{3}}{\left(x - 1\right)^{2} \left(x + \left(x^{3} + x^{2}\right)\right)}\right)^{\frac{4}{5}} \left(x - 1\right)^{4} \left(x^{3} + x^{2} + x\right)^{2}}

  4. Simplificamos:

    (x2−2)2((x−1)(x2+x+1)(6x2(x+2)+x(x2−2)+(x+2)(x2−2))−(x+2)(x2−2)(2x(x2+x+1)+(x−1)(3x2+2x+1)))5x((x+2)(x2−2)3(x−1)2(x2+x+1))45(x−1)3(x2+x+1)2\frac{\left(x^{2} - 2\right)^{2} \left(\left(x - 1\right) \left(x^{2} + x + 1\right) \left(6 x^{2} \left(x + 2\right) + x \left(x^{2} - 2\right) + \left(x + 2\right) \left(x^{2} - 2\right)\right) - \left(x + 2\right) \left(x^{2} - 2\right) \left(2 x \left(x^{2} + x + 1\right) + \left(x - 1\right) \left(3 x^{2} + 2 x + 1\right)\right)\right)}{5 x \left(\frac{\left(x + 2\right) \left(x^{2} - 2\right)^{3}}{\left(x - 1\right)^{2} \left(x^{2} + x + 1\right)}\right)^{\frac{4}{5}} \left(x - 1\right)^{3} \left(x^{2} + x + 1\right)^{2}}

Respuesta:

(x2−2)2((x−1)(x2+x+1)(6x2(x+2)+x(x2−2)+(x+2)(x2−2))−(x+2)(x2−2)(2x(x2+x+1)+(x−1)(3x2+2x+1)))5x((x+2)(x2−2)3(x−1)2(x2+x+1))45(x−1)3(x2+x+1)2\frac{\left(x^{2} - 2\right)^{2} \left(\left(x - 1\right) \left(x^{2} + x + 1\right) \left(6 x^{2} \left(x + 2\right) + x \left(x^{2} - 2\right) + \left(x + 2\right) \left(x^{2} - 2\right)\right) - \left(x + 2\right) \left(x^{2} - 2\right) \left(2 x \left(x^{2} + x + 1\right) + \left(x - 1\right) \left(3 x^{2} + 2 x + 1\right)\right)\right)}{5 x \left(\frac{\left(x + 2\right) \left(x^{2} - 2\right)^{3}}{\left(x - 1\right)^{2} \left(x^{2} + x + 1\right)}\right)^{\frac{4}{5}} \left(x - 1\right)^{3} \left(x^{2} + x + 1\right)^{2}}

Gráfica
02468-8-6-4-2-10105-5
Trazar el gráfico f(x)
Trazar el gráfico f'(x)
Primera derivada [src] 
                                                            /                       /          3           /        3                2\\                                                                               \              
                                                            |          1            |  / 2    \            |/ 2    \       2 / 2    \ ||                                                                               |              
      ____________________________________________          |----------------------*\x*\x  - 2/  + (x + 2)*\\x  - 2/  + 6*x *\x  - 2/ //             3                                                                 |              
     /           3                                          |       2 / 3    2    \                                                          / 2    \          /         2 /             2\              / 3    2    \\|              
    /    / 2    \            1                            2 |(x - 1) *\x  + x  + x/                                                        x*\x  - 2/ *(x + 2)*\- (x - 1) *\1 + 2*x + 3*x / - (-2 + 2*x)*\x  + x  + x//| / 3    2    \
   /   x*\x  - 2/ *----------------------*(x + 2) *(x - 1) *|--------------------------------------------------------------------------- + ----------------------------------------------------------------------------|*\x  + x  + x/
5 /                       2 / 3    2    \                   |                                     5                                                                                         2                          |              
\/                 (x - 1) *\x  + x  + x/                   |                                                                                                                4 / 3    2    \                           |              
                                                            \                                                                                                       5*(x - 1) *\x  + x  + x/                           /              
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                                                                                                                           3                                                                                                          
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                                                                                                         x*(x + 2)*\x  - 2/
x1(x−1)2(x+(x3+x2))(x+2)(x2−2)35(x−1)2(x+(x3+x2))(x(x+2)(x2−2)3(−(x−1)2(3x2+2x+1)−(x+(x3+x2))(2x−2))5(x−1)4(x+(x3+x2))2+1(x−1)2(x+(x3+x2))(x(x2−2)3+(x+2)(6x2(x2−2)2+(x2−2)3))5)x(x+2)(x2−2)3\frac{\sqrt[5]{x \frac{1}{\left(x - 1\right)^{2} \left(x + \left(x^{3} + x^{2}\right)\right)} \left(x + 2\right) \left(x^{2} - 2\right)^{3}} \left(x - 1\right)^{2} \left(x + \left(x^{3} + x^{2}\right)\right) \left(\frac{x \left(x + 2\right) \left(x^{2} - 2\right)^{3} \left(- \left(x - 1\right)^{2} \left(3 x^{2} + 2 x + 1\right) - \left(x + \left(x^{3} + x^{2}\right)\right) \left(2 x - 2\right)\right)}{5 \left(x - 1\right)^{4} \left(x + \left(x^{3} + x^{2}\right)\right)^{2}} + \frac{\frac{1}{\left(x - 1\right)^{2} \left(x + \left(x^{3} + x^{2}\right)\right)} \left(x \left(x^{2} - 2\right)^{3} + \left(x + 2\right) \left(6 x^{2} \left(x^{2} - 2\right)^{2} + \left(x^{2} - 2\right)^{3}\right)\right)}{5}\right)}{x \left(x + 2\right) \left(x^{2} - 2\right)^{3}}
Simplificar
Segunda derivada [src] 
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                                |    |    /         2                                           \   \-2 + x /*|------ + --------------|*\x*\-2 + x / + \-2 + 7*x /*(2 + x)/            2                                                                                                                                2                                                                     2                                                                     2                                                                                        2                                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
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                                |    |  2*\\-2 + x /  + 6*x *\-2 + x / + 3*x*\-6 + 7*x /*(2 + x)/             \         x*\1 + x + x //                                       \-2 + x / *\(-1 + x)*\1 + 2*x + 3*x / + 2*x*\1 + x + x //   6*\-2 + x /*(2 + x)*\(-1 + x)*\1 + 2*x + 3*x / + 2*x*\1 + x + x //   \-2 + x / *(2 + x)*\(-1 + x)*\1 + 2*x + 3*x / + 2*x*\1 + x + x //   4*\-2 + x / *(2 + x)*\(-1 + x)*\1 + 2*x + 3*x / + 2*x*\1 + x + x //   2*\-2 + x / *(2 + x)*\x + x  + x  + (-1 + x) *(1 + 3*x) + 2*(-1 + x)*\1 + 2*x + 3*x //   2*\-2 + x / *(2 + x)*\(-1 + x)*\1 + 2*x + 3*x / + 2*x*\1 + x + x //*\1 + 2*x + 3*x /|     /                                      /      2\         /         /             2\       /         2\\\      /                                      /      2\         /         /             2\       /         2\\\      /                                      /      2\         /         /             2\       /         2\\\     /                                      /      2\         /         /             2\       /         2\\\                      /                                      /      2\         /         /             2\       /         2\\\   /                                      /      2\         /         /             2\       /         2\\\ /                        /      2\                             /                      2\\|
       ________________________ |  5*|- --------------------------------------------------------- + ----------------------------------------------------------------------- + --------------------------------------------------------- + ------------------------------------------------------------------ + ----------------------------------------------------------------- - ------------------------------------------------------------------- + -------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------|     |    /      2\   /        2\           \-2 + x /*(2 + x)*\(-1 + x)*\1 + 2*x + 3*x / + 2*x*\1 + x + x //|      |    /      2\   /        2\           \-2 + x /*(2 + x)*\(-1 + x)*\1 + 2*x + 3*x / + 2*x*\1 + x + x //|      |    /      2\   /        2\           \-2 + x /*(2 + x)*\(-1 + x)*\1 + 2*x + 3*x / + 2*x*\1 + x + x //|     |    /      2\   /        2\           \-2 + x /*(2 + x)*\(-1 + x)*\1 + 2*x + 3*x / + 2*x*\1 + x + x //|     /             2\ |    /      2\   /        2\           \-2 + x /*(2 + x)*\(-1 + x)*\1 + 2*x + 3*x / + 2*x*\1 + x + x //|   |    /      2\   /        2\           \-2 + x /*(2 + x)*\(-1 + x)*\1 + 2*x + 3*x / + 2*x*\1 + x + x //| |      2                 \-2 + x /*(2 + x)   /      2\         |  2      1 + 2*x + 3*x |||
      /            3            |    |                              x                                                                  x                                                                  /         2\                                                   /         2\                                                2          /         2\                                                       2 /         2\                                                                 2 /         2\                                                                                       2                              |   5*|- x*\-2 + x / - \-2 + 7*x /*(2 + x) + ----------------------------------------------------------------|   30*|- x*\-2 + x / - \-2 + 7*x /*(2 + x) + ----------------------------------------------------------------|   10*|- x*\-2 + x / - \-2 + 7*x /*(2 + x) + ----------------------------------------------------------------|   5*|- x*\-2 + x / - \-2 + 7*x /*(2 + x) + ----------------------------------------------------------------|   5*\1 + 2*x + 3*x /*|- x*\-2 + x / - \-2 + 7*x /*(2 + x) + ----------------------------------------------------------------|   |- x*\-2 + x / - \-2 + 7*x /*(2 + x) + ----------------------------------------------------------------|*|-2 + x  + 6*x*(2 + x) + ----------------- - \-2 + x /*(2 + x)*|------ + --------------|||
     /    /      2\             |    |                                                                                                                                                         x*(-1 + x)*\1 + x + x /                                          (-1 + x)*\1 + x + x /                                               x *(-1 + x)*\1 + x + x /                                             x*(-1 + x) *\1 + x + x /                                                       x*(-1 + x) *\1 + x + x /                                                                2          /         2\                               |     |                                                                    /         2\                      |      |                                                                    /         2\                      |      |                                                                    /         2\                      |     |                                                                    /         2\                      |                      |                                                                    /         2\                      |   |                                                                    /         2\                      | |                                x                             |-1 + x     /         2\|||
    /     \-2 + x / *(2 + x)    |    \                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         x *(-1 + x)*\1 + x + x /                               /     \                                                           (-1 + x)*\1 + x + x /                      /      \                                                           (-1 + x)*\1 + x + x /                      /      \                                                           (-1 + x)*\1 + x + x /                      /     \                                                           (-1 + x)*\1 + x + x /                      /                      \                                                           (-1 + x)*\1 + x + x /                      /   \                                                           (-1 + x)*\1 + x + x /                      / \                                                              \         x*\1 + x + x ///|
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5 /             2 /         2\  |                                                                                                                                                                                                                                                                                                      2                                                                                                                                                                                                                                                                                                                                                       2                                                                                                               2                                                                                                      x*(-1 + x)                                                                                                   x*(2 + x)                                                                                                           2 /         2\                                                                                                                                                  /      2\                                                                                                |
\/      (-1 + x) *\1 + x + x /  \                                                                                                                                                                                                                                                                                                -2 + x                                                                                                                                                                                                                                                                                                                                                       x                                                                                                          -2 + x                                                                                                                                                                                                                                                                                                                                       x *\1 + x + x /                                                                                                                                                x*\-2 + x /*(2 + x)                                                                                        /
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                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       25*\-2 + x /*(2 + x)
(x+2)(x2−2)3(x−1)2(x2+x+1)5(30(−x(x2−2)−(x+2)(7x2−2)+(x+2)(x2−2)(2x(x2+x+1)+(x−1)(3x2+2x+1))(x−1)(x2+x+1))x2−2−5(6(x+2)(x2−2)(2x(x2+x+1)+(x−1)(3x2+2x+1))(x−1)(x2+x+1)+(x2−2)(x(x2−2)+(x+2)(7x2−2))(2x−1+3x2+2x+1x(x2+x+1))x−2(6x2(x2−2)+3x(x+2)(7x2−6)+(x2−2)2)x+(x2−2)2(2x(x2+x+1)+(x−1)(3x2+2x+1))x(x−1)(x2+x+1)−4(x+2)(x2−2)2(2x(x2+x+1)+(x−1)(3x2+2x+1))x(x−1)2(x2+x+1)+2(x+2)(x2−2)2(x3+x2+x+(x−1)2(3x+1)+2(x−1)(3x2+2x+1))x(x−1)2(x2+x+1)+(x+2)(x2−2)2(2x(x2+x+1)+(x−1)(3x2+2x+1))x2(x−1)(x2+x+1)−2(x+2)(x2−2)2(2x(x2+x+1)+(x−1)(3x2+2x+1))(3x2+2x+1)x2(x−1)(x2+x+1)2)x2−2+5(−x(x2−2)−(x+2)(7x2−2)+(x+2)(x2−2)(2x(x2+x+1)+(x−1)(3x2+2x+1))(x−1)(x2+x+1))x(x+2)−(−x(x2−2)−(x+2)(7x2−2)+(x+2)(x2−2)(2x(x2+x+1)+(x−1)(3x2+2x+1))(x−1)(x2+x+1))(x2+6x(x+2)−(x+2)(x2−2)(2x−1+3x2+2x+1x(x2+x+1))−2+(x+2)(x2−2)x)x(x+2)(x2−2)−10(−x(x2−2)−(x+2)(7x2−2)+(x+2)(x2−2)(2x(x2+x+1)+(x−1)(3x2+2x+1))(x−1)(x2+x+1))x(x−1)+5(−x(x2−2)−(x+2)(7x2−2)+(x+2)(x2−2)(2x(x2+x+1)+(x−1)(3x2+2x+1))(x−1)(x2+x+1))x2−5(3x2+2x+1)(−x(x2−2)−(x+2)(7x2−2)+(x+2)(x2−2)(2x(x2+x+1)+(x−1)(3x2+2x+1))(x−1)(x2+x+1))x2(x2+x+1))25(x+2)(x2−2)\frac{\sqrt[5]{\frac{\left(x + 2\right) \left(x^{2} - 2\right)^{3}}{\left(x - 1\right)^{2} \left(x^{2} + x + 1\right)}} \left(\frac{30 \left(- x \left(x^{2} - 2\right) - \left(x + 2\right) \left(7 x^{2} - 2\right) + \frac{\left(x + 2\right) \left(x^{2} - 2\right) \left(2 x \left(x^{2} + x + 1\right) + \left(x - 1\right) \left(3 x^{2} + 2 x + 1\right)\right)}{\left(x - 1\right) \left(x^{2} + x + 1\right)}\right)}{x^{2} - 2} - \frac{5 \left(\frac{6 \left(x + 2\right) \left(x^{2} - 2\right) \left(2 x \left(x^{2} + x + 1\right) + \left(x - 1\right) \left(3 x^{2} + 2 x + 1\right)\right)}{\left(x - 1\right) \left(x^{2} + x + 1\right)} + \frac{\left(x^{2} - 2\right) \left(x \left(x^{2} - 2\right) + \left(x + 2\right) \left(7 x^{2} - 2\right)\right) \left(\frac{2}{x - 1} + \frac{3 x^{2} + 2 x + 1}{x \left(x^{2} + x + 1\right)}\right)}{x} - \frac{2 \left(6 x^{2} \left(x^{2} - 2\right) + 3 x \left(x + 2\right) \left(7 x^{2} - 6\right) + \left(x^{2} - 2\right)^{2}\right)}{x} + \frac{\left(x^{2} - 2\right)^{2} \left(2 x \left(x^{2} + x + 1\right) + \left(x - 1\right) \left(3 x^{2} + 2 x + 1\right)\right)}{x \left(x - 1\right) \left(x^{2} + x + 1\right)} - \frac{4 \left(x + 2\right) \left(x^{2} - 2\right)^{2} \left(2 x \left(x^{2} + x + 1\right) + \left(x - 1\right) \left(3 x^{2} + 2 x + 1\right)\right)}{x \left(x - 1\right)^{2} \left(x^{2} + x + 1\right)} + \frac{2 \left(x + 2\right) \left(x^{2} - 2\right)^{2} \left(x^{3} + x^{2} + x + \left(x - 1\right)^{2} \left(3 x + 1\right) + 2 \left(x - 1\right) \left(3 x^{2} + 2 x + 1\right)\right)}{x \left(x - 1\right)^{2} \left(x^{2} + x + 1\right)} + \frac{\left(x + 2\right) \left(x^{2} - 2\right)^{2} \left(2 x \left(x^{2} + x + 1\right) + \left(x - 1\right) \left(3 x^{2} + 2 x + 1\right)\right)}{x^{2} \left(x - 1\right) \left(x^{2} + x + 1\right)} - \frac{2 \left(x + 2\right) \left(x^{2} - 2\right)^{2} \left(2 x \left(x^{2} + x + 1\right) + \left(x - 1\right) \left(3 x^{2} + 2 x + 1\right)\right) \left(3 x^{2} + 2 x + 1\right)}{x^{2} \left(x - 1\right) \left(x^{2} + x + 1\right)^{2}}\right)}{x^{2} - 2} + \frac{5 \left(- x \left(x^{2} - 2\right) - \left(x + 2\right) \left(7 x^{2} - 2\right) + \frac{\left(x + 2\right) \left(x^{2} - 2\right) \left(2 x \left(x^{2} + x + 1\right) + \left(x - 1\right) \left(3 x^{2} + 2 x + 1\right)\right)}{\left(x - 1\right) \left(x^{2} + x + 1\right)}\right)}{x \left(x + 2\right)} - \frac{\left(- x \left(x^{2} - 2\right) - \left(x + 2\right) \left(7 x^{2} - 2\right) + \frac{\left(x + 2\right) \left(x^{2} - 2\right) \left(2 x \left(x^{2} + x + 1\right) + \left(x - 1\right) \left(3 x^{2} + 2 x + 1\right)\right)}{\left(x - 1\right) \left(x^{2} + x + 1\right)}\right) \left(x^{2} + 6 x \left(x + 2\right) - \left(x + 2\right) \left(x^{2} - 2\right) \left(\frac{2}{x - 1} + \frac{3 x^{2} + 2 x + 1}{x \left(x^{2} + x + 1\right)}\right) - 2 + \frac{\left(x + 2\right) \left(x^{2} - 2\right)}{x}\right)}{x \left(x + 2\right) \left(x^{2} - 2\right)} - \frac{10 \left(- x \left(x^{2} - 2\right) - \left(x + 2\right) \left(7 x^{2} - 2\right) + \frac{\left(x + 2\right) \left(x^{2} - 2\right) \left(2 x \left(x^{2} + x + 1\right) + \left(x - 1\right) \left(3 x^{2} + 2 x + 1\right)\right)}{\left(x - 1\right) \left(x^{2} + x + 1\right)}\right)}{x \left(x - 1\right)} + \frac{5 \left(- x \left(x^{2} - 2\right) - \left(x + 2\right) \left(7 x^{2} - 2\right) + \frac{\left(x + 2\right) \left(x^{2} - 2\right) \left(2 x \left(x^{2} + x + 1\right) + \left(x - 1\right) \left(3 x^{2} + 2 x + 1\right)\right)}{\left(x - 1\right) \left(x^{2} + x + 1\right)}\right)}{x^{2}} - \frac{5 \left(3 x^{2} + 2 x + 1\right) \left(- x \left(x^{2} - 2\right) - \left(x + 2\right) \left(7 x^{2} - 2\right) + \frac{\left(x + 2\right) \left(x^{2} - 2\right) \left(2 x \left(x^{2} + x + 1\right) + \left(x - 1\right) \left(3 x^{2} + 2 x + 1\right)\right)}{\left(x - 1\right) \left(x^{2} + x + 1\right)}\right)}{x^{2} \left(x^{2} + x + 1\right)}\right)}{25 \left(x + 2\right) \left(x^{2} - 2\right)}
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Gráfico
Derivada de ((x*(x^2-2)^3*(x+2))/((x-1)^2*(x^3+x^2+x)))^1/5
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