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(x(x+1)(x-2))^(1/3)/((x²+1)(2x+3))^(1/3)
Derivada de la función/ (x(x+1)(x-2))^(1/3)/((x²+1)(2x+3))^(1/3)

Derivada de (x(x+1)(x-2))^(1/3)/((x²+1)(2x+3))^(1/3)

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

Gráfico:

interior superior

Definida a trozos:

Solución

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

  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)(x+1)3f{\left(x \right)} = \sqrt[3]{x \left(x - 2\right) \left(x + 1\right)} y g(x)=2x+33x2+13g{\left(x \right)} = \sqrt[3]{2 x + 3} \sqrt[3]{x^{2} + 1}.

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

    1. Sustituimos u=x(x−2)(x+1)u = x \left(x - 2\right) \left(x + 1\right).

    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(x−2)(x+1)\frac{d}{d x} x \left(x - 2\right) \left(x + 1\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)=x+1g{\left(x \right)} = x + 1; calculamos ddxg(x)\frac{d}{d x} g{\left(x \right)}:

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

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

          2. Según el principio, aplicamos: xx tenemos 11

          Como resultado de: 11

        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 −2-2 es igual a cero.

          2. Según el principio, aplicamos: xx tenemos 11

          Como resultado de: 11

        Como resultado de: x(x−2)+x(x+1)+(x−2)(x+1)x \left(x - 2\right) + x \left(x + 1\right) + \left(x - 2\right) \left(x + 1\right)

      Como resultado de la secuencia de reglas:

      x(x−2)+x(x+1)+(x−2)(x+1)3(x(x−2)(x+1))23\frac{x \left(x - 2\right) + x \left(x + 1\right) + \left(x - 2\right) \left(x + 1\right)}{3 \left(x \left(x - 2\right) \left(x + 1\right)\right)^{\frac{2}{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)=x2+13f{\left(x \right)} = \sqrt[3]{x^{2} + 1}; calculamos ddxf(x)\frac{d}{d x} f{\left(x \right)}:

      1. Sustituimos u=x2+1u = x^{2} + 1.

      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 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 11 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:

        2x3(x2+1)23\frac{2 x}{3 \left(x^{2} + 1\right)^{\frac{2}{3}}}

      g(x)=2x+33g{\left(x \right)} = \sqrt[3]{2 x + 3}; calculamos ddxg(x)\frac{d}{d x} g{\left(x \right)}:

      1. Sustituimos u=2x+3u = 2 x + 3.

      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 ddx(2x+3)\frac{d}{d x} \left(2 x + 3\right):

        1. diferenciamos 2x+32 x + 3 miembro por miembro:

          1. La derivada de una constante 33 es igual a cero.

          2. La derivada del producto de una constante por función es igual al producto de esta constante por la derivada de esta función.

            1. Según el principio, aplicamos: xx tenemos 11

            Entonces, como resultado: 22

          Como resultado de: 22

        Como resultado de la secuencia de reglas:

        23(2x+3)23\frac{2}{3 \left(2 x + 3\right)^{\frac{2}{3}}}

      Como resultado de: 2x2x+333(x2+1)23+2x2+133(2x+3)23\frac{2 x \sqrt[3]{2 x + 3}}{3 \left(x^{2} + 1\right)^{\frac{2}{3}}} + \frac{2 \sqrt[3]{x^{2} + 1}}{3 \left(2 x + 3\right)^{\frac{2}{3}}}

    Ahora aplicamos la regla de la derivada de una divesión:

    −x(x−2)(x+1)3(2x2x+333(x2+1)23+2x2+133(2x+3)23)+2x+33x2+13(x(x−2)+x(x+1)+(x−2)(x+1))3(x(x−2)(x+1))23(2x+3)23(x2+1)23\frac{- \sqrt[3]{x \left(x - 2\right) \left(x + 1\right)} \left(\frac{2 x \sqrt[3]{2 x + 3}}{3 \left(x^{2} + 1\right)^{\frac{2}{3}}} + \frac{2 \sqrt[3]{x^{2} + 1}}{3 \left(2 x + 3\right)^{\frac{2}{3}}}\right) + \frac{\sqrt[3]{2 x + 3} \sqrt[3]{x^{2} + 1} \left(x \left(x - 2\right) + x \left(x + 1\right) + \left(x - 2\right) \left(x + 1\right)\right)}{3 \left(x \left(x - 2\right) \left(x + 1\right)\right)^{\frac{2}{3}}}}{\left(2 x + 3\right)^{\frac{2}{3}} \left(x^{2} + 1\right)^{\frac{2}{3}}}

  2. Simplificamos:

    5x4+12x3+13x2−6x−63(x(x2−x−2))232x+33x2+13(2x3+3x2+2x+3)\frac{5 x^{4} + 12 x^{3} + 13 x^{2} - 6 x - 6}{3 \left(x \left(x^{2} - x - 2\right)\right)^{\frac{2}{3}} \sqrt[3]{2 x + 3} \sqrt[3]{x^{2} + 1} \left(2 x^{3} + 3 x^{2} + 2 x + 3\right)}

Respuesta:

5x4+12x3+13x2−6x−63(x(x2−x−2))232x+33x2+13(2x3+3x2+2x+3)\frac{5 x^{4} + 12 x^{3} + 13 x^{2} - 6 x - 6}{3 \left(x \left(x^{2} - x - 2\right)\right)^{\frac{2}{3}} \sqrt[3]{2 x + 3} \sqrt[3]{x^{2} + 1} \left(2 x^{3} + 3 x^{2} + 2 x + 3\right)}

Gráfica
02468-8-6-4-2-1010-50100
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Primera derivada [src] 
                                                     3 ___________________            1            /x*(x + 1)   (1 + 2*x)*(x - 2)\
                        /       2                \   \/ x*(x + 1)*(x - 2) *-----------------------*|--------- + -----------------|
  3 ___________________ |2   2*x    2*x*(2*x + 3)|                            ________             \    3               3        /
  \/ x*(x + 1)*(x - 2) *|- + ---- + -------------|                         3 /  2      3 _________                                
                        \3    3           3      /                         \/  x  + 1 *\/ 2*x + 3                                 
- ------------------------------------------------ + -----------------------------------------------------------------------------
                      4/3                                                          x*(x + 1)*(x - 2)                              
              / 2    \             4/3                                                                                            
              \x  + 1/   *(2*x + 3)
−x(x−2)(x+1)3(2x23+2x(2x+3)3+23)(2x+3)43(x2+1)43+12x+33x2+13x(x−2)(x+1)3(x(x+1)3+(x−2)(2x+1)3)x(x−2)(x+1)- \frac{\sqrt[3]{x \left(x - 2\right) \left(x + 1\right)} \left(\frac{2 x^{2}}{3} + \frac{2 x \left(2 x + 3\right)}{3} + \frac{2}{3}\right)}{\left(2 x + 3\right)^{\frac{4}{3}} \left(x^{2} + 1\right)^{\frac{4}{3}}} + \frac{\frac{1}{\sqrt[3]{2 x + 3} \sqrt[3]{x^{2} + 1}} \sqrt[3]{x \left(x - 2\right) \left(x + 1\right)} \left(\frac{x \left(x + 1\right)}{3} + \frac{\left(x - 2\right) \left(2 x + 1\right)}{3}\right)}{x \left(x - 2\right) \left(x + 1\right)}
Simplificar
Segunda derivada [src] 
                       /  /                                                            /     2              \       /     2              \\                                                                                                                                                                                                                                                                     \
                       |  |              /   1        x   \ /     2              \   6*\1 + x  + x*(3 + 2*x)/   6*x*\1 + x  + x*(3 + 2*x)/|                                                                                                                                                                                                                                                                     |
                       |2*|-9 - 18*x + 2*|------- + ------|*\1 + x  + x*(3 + 2*x)/ + ------------------------ + --------------------------|              3*(x*(1 + x) + (1 + 2*x)*(-2 + x))   3*(x*(1 + x) + (1 + 2*x)*(-2 + x))   3*(x*(1 + x) + (1 + 2*x)*(-2 + x))   (x*(1 + x) + (1 + 2*x)*(-2 + x))*(x*(1 + x) + x*(-2 + x) + (1 + x)*(-2 + x))                                                            |
                       |  |              |3 + 2*x        2|                                  3 + 2*x                           2          |   6 - 18*x + ---------------------------------- + ---------------------------------- + ---------------------------------- - ----------------------------------------------------------------------------                                      /     2              \|
3 ____________________ |  \              \          1 + x /                                                               1 + x           /                              x                                  1 + x                                -2 + x                                              x*(1 + x)*(-2 + x)                                4*(x*(1 + x) + (1 + 2*x)*(-2 + x))*\1 + x  + x*(3 + 2*x)/|
\/ x*(1 + x)*(-2 + x) *|------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ - ---------------------------------------------------------|
                       |                                                 /     2\                                                                                                                                                       x*(1 + x)*(-2 + x)                                                                                                                 /     2\                             |
                       \                                                 \1 + x /*(3 + 2*x)                                                                                                                                                                                                                                                                      x*(1 + x)*\1 + x /*(-2 + x)*(3 + 2*x)          /
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                                                                                                                                                                                              3 /      2  3 _________                                                                                                                                                                                            
                                                                                                                                                                                            9*\/  1 + x  *\/ 3 + 2*x
x(x−2)(x+1)3(2(−18x+6x(x2+x(2x+3)+1)x2+1+2(xx2+1+12x+3)(x2+x(2x+3)+1)−9+6(x2+x(2x+3)+1)2x+3)(2x+3)(x2+1)−−18x+6+3(x(x+1)+(x−2)(2x+1))x+1+3(x(x+1)+(x−2)(2x+1))x−2+3(x(x+1)+(x−2)(2x+1))x−(x(x+1)+(x−2)(2x+1))(x(x−2)+x(x+1)+(x−2)(x+1))x(x−2)(x+1)x(x−2)(x+1)−4(x(x+1)+(x−2)(2x+1))(x2+x(2x+3)+1)x(x−2)(x+1)(2x+3)(x2+1))92x+33x2+13\frac{\sqrt[3]{x \left(x - 2\right) \left(x + 1\right)} \left(\frac{2 \left(- 18 x + \frac{6 x \left(x^{2} + x \left(2 x + 3\right) + 1\right)}{x^{2} + 1} + 2 \left(\frac{x}{x^{2} + 1} + \frac{1}{2 x + 3}\right) \left(x^{2} + x \left(2 x + 3\right) + 1\right) - 9 + \frac{6 \left(x^{2} + x \left(2 x + 3\right) + 1\right)}{2 x + 3}\right)}{\left(2 x + 3\right) \left(x^{2} + 1\right)} - \frac{- 18 x + 6 + \frac{3 \left(x \left(x + 1\right) + \left(x - 2\right) \left(2 x + 1\right)\right)}{x + 1} + \frac{3 \left(x \left(x + 1\right) + \left(x - 2\right) \left(2 x + 1\right)\right)}{x - 2} + \frac{3 \left(x \left(x + 1\right) + \left(x - 2\right) \left(2 x + 1\right)\right)}{x} - \frac{\left(x \left(x + 1\right) + \left(x - 2\right) \left(2 x + 1\right)\right) \left(x \left(x - 2\right) + x \left(x + 1\right) + \left(x - 2\right) \left(x + 1\right)\right)}{x \left(x - 2\right) \left(x + 1\right)}}{x \left(x - 2\right) \left(x + 1\right)} - \frac{4 \left(x \left(x + 1\right) + \left(x - 2\right) \left(2 x + 1\right)\right) \left(x^{2} + x \left(2 x + 3\right) + 1\right)}{x \left(x - 2\right) \left(x + 1\right) \left(2 x + 3\right) \left(x^{2} + 1\right)}\right)}{9 \sqrt[3]{2 x + 3} \sqrt[3]{x^{2} + 1}}
Simplificar
Tercera derivada [src] 
                       /    /                                                                                                                                                                                                                 /   1        x   \ /     2              \                                      /   1        x   \ /     2              \                              \                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   \
                       |    |                                                                                                                                                                                                               6*|------- + ------|*\1 + x  + x*(3 + 2*x)/                                  6*x*|------- + ------|*\1 + x  + x*(3 + 2*x)/                              |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
                       |    |                            /                              2                       \                    /     2              \                                       /     2              \                      |3 + 2*x        2|                              2 /     2              \       |3 + 2*x        2|                               /     2              \|                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
                       |    |     /     2              \ |    3          8           8*x             4*x        |   63*(1 + 2*x)   9*\1 + x  + x*(3 + 2*x)/               /   1        x   \   42*\1 + x  + x*(3 + 2*x)/   63*x*(1 + 2*x)     \          1 + x /                          42*x *\1 + x  + x*(3 + 2*x)/       \          1 + x /                          48*x*\1 + x  + x*(3 + 2*x)/|                                                                                                                                                                                                                                                                                                                                       2                                                                                                                                                                                                                                                                                                                                                                                                                                      /                                                            /     2              \       /     2              \\                                                                                                                                                                                                                                    |
                       |  4*|27 + \1 + x  + x*(3 + 2*x)/*|- ------ + ---------- + --------- + ------------------| - ------------ - ------------------------ - 9*(1 + 2*x)*|------- + ------| + ------------------------- - -------------- + ------------------------------------------- + ---------------------------- + --------------------------------------------- + ---------------------------|        36*(-1 + 3*x)   36*(-1 + 3*x)   36*(-1 + 3*x)   18*(x*(1 + x) + (1 + 2*x)*(-2 + x))   18*(x*(1 + x) + (1 + 2*x)*(-2 + x))   18*(x*(1 + x) + (1 + 2*x)*(-2 + x))   18*(x*(1 + x) + (1 + 2*x)*(-2 + x))   18*(x*(1 + x) + (1 + 2*x)*(-2 + x))   18*(x*(1 + x) + (1 + 2*x)*(-2 + x))   (x*(1 + x) + x*(-2 + x) + (1 + x)*(-2 + x)) *(x*(1 + x) + (1 + 2*x)*(-2 + x))   9*(x*(1 + x) + (1 + 2*x)*(-2 + x))*(x*(1 + x) + x*(-2 + x) + (1 + x)*(-2 + x))   9*(x*(1 + x) + (1 + 2*x)*(-2 + x))*(x*(1 + x) + x*(-2 + x) + (1 + x)*(-2 + x))   9*(x*(1 + x) + (1 + 2*x)*(-2 + x))*(x*(1 + x) + x*(-2 + x) + (1 + x)*(-2 + x))   6*(-1 + 3*x)*(x*(1 + x) + (1 + 2*x)*(-2 + x))   12*(-1 + 3*x)*(x*(1 + x) + x*(-2 + x) + (1 + x)*(-2 + x))                                      |              /   1        x   \ /     2              \   6*\1 + x  + x*(3 + 2*x)/   6*x*\1 + x  + x*(3 + 2*x)/|                                                                                                                                                                                                                                    |
                       |    |                            |       2            2           2   /     2\          |     3 + 2*x                    2                        |3 + 2*x        2|                    2                   2                         3 + 2*x                                      2                                      2                           /     2\              |   54 - ------------- - ------------- - ------------- + ----------------------------------- + ----------------------------------- + ----------------------------------- + ----------------------------------- + ----------------------------------- + ----------------------------------- + ----------------------------------------------------------------------------- - ------------------------------------------------------------------------------ - ------------------------------------------------------------------------------ - ------------------------------------------------------------------------------ + --------------------------------------------- + ---------------------------------------------------------   6*(x*(1 + x) + (1 + 2*x)*(-2 + x))*|-9 - 18*x + 2*|------- + ------|*\1 + x  + x*(3 + 2*x)/ + ------------------------ + --------------------------|     /     2              \ /           3*(x*(1 + x) + (1 + 2*x)*(-2 + x))   3*(x*(1 + x) + (1 + 2*x)*(-2 + x))   3*(x*(1 + x) + (1 + 2*x)*(-2 + x))   (x*(1 + x) + (1 + 2*x)*(-2 + x))*(x*(1 + x) + x*(-2 + x) + (1 + x)*(-2 + x))\|
                       |    |                            |  1 + x    (3 + 2*x)    /     2\    \1 + x /*(3 + 2*x)|                           1 + x                         \          1 + x /           (3 + 2*x)               1 + x                                                               /     2\                                  1 + x                            \1 + x /*(3 + 2*x)    |              x             1 + x           -2 + x                        2                                        2                                     2                             x*(1 + x)                             x*(-2 + x)                         (1 + x)*(-2 + x)                                         2        2         2                                                                              2                                                                       2                                                                        2                                                                x*(1 + x)*(-2 + x)                                    x*(1 + x)*(-2 + x)                                                         |              |3 + 2*x        2|                                  3 + 2*x                           2          |   6*\1 + x  + x*(3 + 2*x)/*|6 - 18*x + ---------------------------------- + ---------------------------------- + ---------------------------------- - ----------------------------------------------------------------------------||
3 ____________________ |    \                            \                        \1 + x /                      /                                                                                                                                                                                  \1 + x /                                                                                         /                                                                         x                                  (1 + x)                              (-2 + x)                                                                                                                                                               x *(1 + x) *(-2 + x)                                                             x*(1 + x)*(-2 + x)                                                               x*(1 + x) *(-2 + x)                                                              x *(1 + x)*(-2 + x)                                                                                                                                                                                \              \          1 + x /                                                               1 + x           /                            \                           x                                  1 + x                                -2 + x                                              x*(1 + x)*(-2 + x)                             /|
\/ x*(1 + x)*(-2 + x) *|- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|
                       |                                                                                                                                                                               /     2\                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   x*(1 + x)*(-2 + x)                                                                                                                                                                                                                                                                                                                                                                                                                             /     2\                                                                                                                                                                                      /     2\                                                                                                                 |
                       \                                                                                                                                                                               \1 + x /*(3 + 2*x)                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              x*(1 + x)*\1 + x /*(-2 + x)*(3 + 2*x)                                                                                                                                                         x*(1 + x)*\1 + x /*(-2 + x)*(3 + 2*x)                                                                                              /
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                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           3 /      2  3 _________                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        27*\/  1 + x  *\/ 3 + 2*x
x(x−2)(x+1)3(−4(42x2(x2+x(2x+3)+1)(x2+1)2−63x(2x+1)x2+1+6x(xx2+1+12x+3)(x2+x(2x+3)+1)x2+1+48x(x2+x(2x+3)+1)(2x+3)(x2+1)−9(2x+1)(xx2+1+12x+3)−63(2x+1)2x+3+(x2+x(2x+3)+1)(8x2(x2+1)2+4x(2x+3)(x2+1)−3x2+1+8(2x+3)2)+27−9(x2+x(2x+3)+1)x2+1+6(xx2+1+12x+3)(x2+x(2x+3)+1)2x+3+42(x2+x(2x+3)+1)(2x+3)2)(2x+3)(x2+1)+54−36(3x−1)x+1+18(x(x+1)+(x−2)(2x+1))(x+1)2−36(3x−1)x−2+18(x(x+1)+(x−2)(2x+1))(x−2)(x+1)+18(x(x+1)+(x−2)(2x+1))(x−2)2−36(3x−1)x+18(x(x+1)+(x−2)(2x+1))x(x+1)+18(x(x+1)+(x−2)(2x+1))x(x−2)+6(3x−1)(x(x+1)+(x−2)(2x+1))x(x−2)(x+1)+12(3x−1)(x(x−2)+x(x+1)+(x−2)(x+1))x(x−2)(x+1)−9(x(x+1)+(x−2)(2x+1))(x(x−2)+x(x+1)+(x−2)(x+1))x(x−2)(x+1)2−9(x(x+1)+(x−2)(2x+1))(x(x−2)+x(x+1)+(x−2)(x+1))x(x−2)2(x+1)+18(x(x+1)+(x−2)(2x+1))x2−9(x(x+1)+(x−2)(2x+1))(x(x−2)+x(x+1)+(x−2)(x+1))x2(x−2)(x+1)+(x(x+1)+(x−2)(2x+1))(x(x−2)+x(x+1)+(x−2)(x+1))2x2(x−2)2(x+1)2x(x−2)(x+1)+6(x(x+1)+(x−2)(2x+1))(−18x+6x(x2+x(2x+3)+1)x2+1+2(xx2+1+12x+3)(x2+x(2x+3)+1)−9+6(x2+x(2x+3)+1)2x+3)x(x−2)(x+1)(2x+3)(x2+1)+6(x2+x(2x+3)+1)(−18x+6+3(x(x+1)+(x−2)(2x+1))x+1+3(x(x+1)+(x−2)(2x+1))x−2+3(x(x+1)+(x−2)(2x+1))x−(x(x+1)+(x−2)(2x+1))(x(x−2)+x(x+1)+(x−2)(x+1))x(x−2)(x+1))x(x−2)(x+1)(2x+3)(x2+1))272x+33x2+13\frac{\sqrt[3]{x \left(x - 2\right) \left(x + 1\right)} \left(- \frac{4 \left(\frac{42 x^{2} \left(x^{2} + x \left(2 x + 3\right) + 1\right)}{\left(x^{2} + 1\right)^{2}} - \frac{63 x \left(2 x + 1\right)}{x^{2} + 1} + \frac{6 x \left(\frac{x}{x^{2} + 1} + \frac{1}{2 x + 3}\right) \left(x^{2} + x \left(2 x + 3\right) + 1\right)}{x^{2} + 1} + \frac{48 x \left(x^{2} + x \left(2 x + 3\right) + 1\right)}{\left(2 x + 3\right) \left(x^{2} + 1\right)} - 9 \left(2 x + 1\right) \left(\frac{x}{x^{2} + 1} + \frac{1}{2 x + 3}\right) - \frac{63 \left(2 x + 1\right)}{2 x + 3} + \left(x^{2} + x \left(2 x + 3\right) + 1\right) \left(\frac{8 x^{2}}{\left(x^{2} + 1\right)^{2}} + \frac{4 x}{\left(2 x + 3\right) \left(x^{2} + 1\right)} - \frac{3}{x^{2} + 1} + \frac{8}{\left(2 x + 3\right)^{2}}\right) + 27 - \frac{9 \left(x^{2} + x \left(2 x + 3\right) + 1\right)}{x^{2} + 1} + \frac{6 \left(\frac{x}{x^{2} + 1} + \frac{1}{2 x + 3}\right) \left(x^{2} + x \left(2 x + 3\right) + 1\right)}{2 x + 3} + \frac{42 \left(x^{2} + x \left(2 x + 3\right) + 1\right)}{\left(2 x + 3\right)^{2}}\right)}{\left(2 x + 3\right) \left(x^{2} + 1\right)} + \frac{54 - \frac{36 \left(3 x - 1\right)}{x + 1} + \frac{18 \left(x \left(x + 1\right) + \left(x - 2\right) \left(2 x + 1\right)\right)}{\left(x + 1\right)^{2}} - \frac{36 \left(3 x - 1\right)}{x - 2} + \frac{18 \left(x \left(x + 1\right) + \left(x - 2\right) \left(2 x + 1\right)\right)}{\left(x - 2\right) \left(x + 1\right)} + \frac{18 \left(x \left(x + 1\right) + \left(x - 2\right) \left(2 x + 1\right)\right)}{\left(x - 2\right)^{2}} - \frac{36 \left(3 x - 1\right)}{x} + \frac{18 \left(x \left(x + 1\right) + \left(x - 2\right) \left(2 x + 1\right)\right)}{x \left(x + 1\right)} + \frac{18 \left(x \left(x + 1\right) + \left(x - 2\right) \left(2 x + 1\right)\right)}{x \left(x - 2\right)} + \frac{6 \left(3 x - 1\right) \left(x \left(x + 1\right) + \left(x - 2\right) \left(2 x + 1\right)\right)}{x \left(x - 2\right) \left(x + 1\right)} + \frac{12 \left(3 x - 1\right) \left(x \left(x - 2\right) + x \left(x + 1\right) + \left(x - 2\right) \left(x + 1\right)\right)}{x \left(x - 2\right) \left(x + 1\right)} - \frac{9 \left(x \left(x + 1\right) + \left(x - 2\right) \left(2 x + 1\right)\right) \left(x \left(x - 2\right) + x \left(x + 1\right) + \left(x - 2\right) \left(x + 1\right)\right)}{x \left(x - 2\right) \left(x + 1\right)^{2}} - \frac{9 \left(x \left(x + 1\right) + \left(x - 2\right) \left(2 x + 1\right)\right) \left(x \left(x - 2\right) + x \left(x + 1\right) + \left(x - 2\right) \left(x + 1\right)\right)}{x \left(x - 2\right)^{2} \left(x + 1\right)} + \frac{18 \left(x \left(x + 1\right) + \left(x - 2\right) \left(2 x + 1\right)\right)}{x^{2}} - \frac{9 \left(x \left(x + 1\right) + \left(x - 2\right) \left(2 x + 1\right)\right) \left(x \left(x - 2\right) + x \left(x + 1\right) + \left(x - 2\right) \left(x + 1\right)\right)}{x^{2} \left(x - 2\right) \left(x + 1\right)} + \frac{\left(x \left(x + 1\right) + \left(x - 2\right) \left(2 x + 1\right)\right) \left(x \left(x - 2\right) + x \left(x + 1\right) + \left(x - 2\right) \left(x + 1\right)\right)^{2}}{x^{2} \left(x - 2\right)^{2} \left(x + 1\right)^{2}}}{x \left(x - 2\right) \left(x + 1\right)} + \frac{6 \left(x \left(x + 1\right) + \left(x - 2\right) \left(2 x + 1\right)\right) \left(- 18 x + \frac{6 x \left(x^{2} + x \left(2 x + 3\right) + 1\right)}{x^{2} + 1} + 2 \left(\frac{x}{x^{2} + 1} + \frac{1}{2 x + 3}\right) \left(x^{2} + x \left(2 x + 3\right) + 1\right) - 9 + \frac{6 \left(x^{2} + x \left(2 x + 3\right) + 1\right)}{2 x + 3}\right)}{x \left(x - 2\right) \left(x + 1\right) \left(2 x + 3\right) \left(x^{2} + 1\right)} + \frac{6 \left(x^{2} + x \left(2 x + 3\right) + 1\right) \left(- 18 x + 6 + \frac{3 \left(x \left(x + 1\right) + \left(x - 2\right) \left(2 x + 1\right)\right)}{x + 1} + \frac{3 \left(x \left(x + 1\right) + \left(x - 2\right) \left(2 x + 1\right)\right)}{x - 2} + \frac{3 \left(x \left(x + 1\right) + \left(x - 2\right) \left(2 x + 1\right)\right)}{x} - \frac{\left(x \left(x + 1\right) + \left(x - 2\right) \left(2 x + 1\right)\right) \left(x \left(x - 2\right) + x \left(x + 1\right) + \left(x - 2\right) \left(x + 1\right)\right)}{x \left(x - 2\right) \left(x + 1\right)}\right)}{x \left(x - 2\right) \left(x + 1\right) \left(2 x + 3\right) \left(x^{2} + 1\right)}\right)}{27 \sqrt[3]{2 x + 3} \sqrt[3]{x^{2} + 1}}
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Gráfico
Derivada de (x(x+1)(x-2))^(1/3)/((x²+1)(2x+3))^(1/3)
    © Sr Examen – Калькулятор