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Derivada de (x(x-2))/((x+1)^(2/3)*(x-2)^(4/3))

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

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Definida a trozos:

Solución

Ha introducido [src]
      x*(x - 2)      
---------------------
       2/3        4/3
(x + 1)   *(x - 2)   
x(x2)(x2)43(x+1)23\frac{x \left(x - 2\right)}{\left(x - 2\right)^{\frac{4}{3}} \left(x + 1\right)^{\frac{2}{3}}}
(x*(x - 2))/(((x + 1)^(2/3)*(x - 2)^(4/3)))
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(x2)f{\left(x \right)} = x \left(x - 2\right) y g(x)=(x2)43(x+1)23g{\left(x \right)} = \left(x - 2\right)^{\frac{4}{3}} \left(x + 1\right)^{\frac{2}{3}}.

    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)=x2g{\left(x \right)} = x - 2; calculamos ddxg(x)\frac{d}{d x} g{\left(x \right)}:

      1. diferenciamos x2x - 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: 2x22 x - 2

    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)23f{\left(x \right)} = \left(x + 1\right)^{\frac{2}{3}}; calculamos ddxf(x)\frac{d}{d x} f{\left(x \right)}:

      1. Sustituimos u=x+1u = x + 1.

      2. Según el principio, aplicamos: u23u^{\frac{2}{3}} tenemos 23u3\frac{2}{3 \sqrt[3]{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 11 es igual a cero.

          2. Según el principio, aplicamos: xx tenemos 11

          Como resultado de: 11

        Como resultado de la secuencia de reglas:

        23x+13\frac{2}{3 \sqrt[3]{x + 1}}

      g(x)=(x2)43g{\left(x \right)} = \left(x - 2\right)^{\frac{4}{3}}; calculamos ddxg(x)\frac{d}{d x} g{\left(x \right)}:

      1. Sustituimos u=x2u = x - 2.

      2. Según el principio, aplicamos: u43u^{\frac{4}{3}} tenemos 4u33\frac{4 \sqrt[3]{u}}{3}

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

        1. diferenciamos x2x - 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 la secuencia de reglas:

        4x233\frac{4 \sqrt[3]{x - 2}}{3}

      Como resultado de: 2(x2)433x+13+4x23(x+1)233\frac{2 \left(x - 2\right)^{\frac{4}{3}}}{3 \sqrt[3]{x + 1}} + \frac{4 \sqrt[3]{x - 2} \left(x + 1\right)^{\frac{2}{3}}}{3}

    Ahora aplicamos la regla de la derivada de una divesión:

    x(x2)(2(x2)433x+13+4x23(x+1)233)+(x2)43(x+1)23(2x2)(x2)83(x+1)43\frac{- x \left(x - 2\right) \left(\frac{2 \left(x - 2\right)^{\frac{4}{3}}}{3 \sqrt[3]{x + 1}} + \frac{4 \sqrt[3]{x - 2} \left(x + 1\right)^{\frac{2}{3}}}{3}\right) + \left(x - 2\right)^{\frac{4}{3}} \left(x + 1\right)^{\frac{2}{3}} \left(2 x - 2\right)}{\left(x - 2\right)^{\frac{8}{3}} \left(x + 1\right)^{\frac{4}{3}}}

  2. Simplificamos:

    2x23(x+1)23(x2+x+2)\frac{2}{\sqrt[3]{x - 2} \left(x + 1\right)^{\frac{2}{3}} \left(- x^{2} + x + 2\right)}


Respuesta:

2x23(x+1)23(x2+x+2)\frac{2}{\sqrt[3]{x - 2} \left(x + 1\right)^{\frac{2}{3}} \left(- x^{2} + x + 2\right)}

Gráfica
02468-8-6-4-2-1010-2525
Primera derivada [src]
                                     /           2/3 3 _______            4/3\
                                     |  4*(x + 1)   *\/ x - 2    2*(x - 2)   |
                                   x*|- ---------------------- - ------------|
                                     |            3                3 _______ |
          1                          \                           3*\/ x + 1  /
---------------------*(-2 + 2*x) + -------------------------------------------
       2/3        4/3                                4/3        5/3           
(x + 1)   *(x - 2)                            (x + 1)   *(x - 2)              
x(2(x2)433x+134x23(x+1)233)(x2)53(x+1)43+1(x2)43(x+1)23(2x2)\frac{x \left(- \frac{2 \left(x - 2\right)^{\frac{4}{3}}}{3 \sqrt[3]{x + 1}} - \frac{4 \sqrt[3]{x - 2} \left(x + 1\right)^{\frac{2}{3}}}{3}\right)}{\left(x - 2\right)^{\frac{5}{3}} \left(x + 1\right)^{\frac{4}{3}}} + \frac{1}{\left(x - 2\right)^{\frac{4}{3}} \left(x + 1\right)^{\frac{2}{3}}} \left(2 x - 2\right)
Segunda derivada [src]
  /      /            4/3            2/3     3 ________                                                                                                              \                                        \
  |      |    (-2 + x)      2*(1 + x)      8*\/ -2 + x      /         2/3     -2 + x \                                                     /         2/3     -2 + x \|                                        |
  |      |  - ----------- + ------------ + ------------   2*|2*(1 + x)    + ---------|                                                   4*|2*(1 + x)    + ---------||                                        |
  |      |            4/3           2/3     3 _______       |               3 _______|                                                     |               3 _______||                                        |
  |      |     (1 + x)      (-2 + x)        \/ 1 + x        \               \/ 1 + x /     /  1       2   \ /         2/3     -2 + x \     \               \/ 1 + x /|              /         2/3     -2 + x \|
  |    x*|- ------------------------------------------- + ---------------------------- + 2*|----- + ------|*|2*(1 + x)    + ---------| + ----------------------------|   4*(-1 + x)*|2*(1 + x)    + ---------||
  |      |                   3 ________                              1 + x                 \1 + x   -2 + x/ |               3 _______|              -2 + x           |              |               3 _______||
  |      \                   \/ -2 + x                                                                      \               \/ 1 + x /                               /              \               \/ 1 + x /|
2*|1 + --------------------------------------------------------------------------------------------------------------------------------------------------------------- - -------------------------------------|
  |                                                                                       2/3                                                                                             2/3                 |
  \                                                                              9*(1 + x)                                                                                       3*(1 + x)   *(-2 + x)        /
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                    2/3         4/3                                                                                            
                                                                                             (1 + x)   *(-2 + x)                                                                                               
2(x(2(x2x+13+2(x+1)23)(1x+1+2x2)+2(x2x+13+2(x+1)23)x+1+4(x2x+13+2(x+1)23)x2(x2)43(x+1)43+8x23x+13+2(x+1)23(x2)23x23)9(x+1)23+14(x1)(x2x+13+2(x+1)23)3(x2)(x+1)23)(x2)43(x+1)23\frac{2 \left(\frac{x \left(2 \left(\frac{x - 2}{\sqrt[3]{x + 1}} + 2 \left(x + 1\right)^{\frac{2}{3}}\right) \left(\frac{1}{x + 1} + \frac{2}{x - 2}\right) + \frac{2 \left(\frac{x - 2}{\sqrt[3]{x + 1}} + 2 \left(x + 1\right)^{\frac{2}{3}}\right)}{x + 1} + \frac{4 \left(\frac{x - 2}{\sqrt[3]{x + 1}} + 2 \left(x + 1\right)^{\frac{2}{3}}\right)}{x - 2} - \frac{- \frac{\left(x - 2\right)^{\frac{4}{3}}}{\left(x + 1\right)^{\frac{4}{3}}} + \frac{8 \sqrt[3]{x - 2}}{\sqrt[3]{x + 1}} + \frac{2 \left(x + 1\right)^{\frac{2}{3}}}{\left(x - 2\right)^{\frac{2}{3}}}}{\sqrt[3]{x - 2}}\right)}{9 \left(x + 1\right)^{\frac{2}{3}}} + 1 - \frac{4 \left(x - 1\right) \left(\frac{x - 2}{\sqrt[3]{x + 1}} + 2 \left(x + 1\right)^{\frac{2}{3}}\right)}{3 \left(x - 2\right) \left(x + 1\right)^{\frac{2}{3}}}\right)}{\left(x - 2\right)^{\frac{4}{3}} \left(x + 1\right)^{\frac{2}{3}}}
Tercera derivada [src]
  /                               /                                                                         /          4/3            2/3     3 ________\     /        4/3           2/3     3 ________                        \                                                                                   /          4/3            2/3     3 ________\     /          4/3            2/3     3 ________\                                                                                                                                \            /            4/3            2/3     3 ________                                                                                                              \\
  |                               |                                                                         |  (-2 + x)      2*(1 + x)      8*\/ -2 + x |     |(-2 + x)       (1 + x)      3*\/ -2 + x              3          |     /         2/3     -2 + x \      /         2/3     -2 + x \   /  1       2   \ |  (-2 + x)      2*(1 + x)      8*\/ -2 + x |     |  (-2 + x)      2*(1 + x)      8*\/ -2 + x |     /  1       2   \ /         2/3     -2 + x \     /  1       2   \ /         2/3     -2 + x \      /         2/3     -2 + x \|            |    (-2 + x)      2*(1 + x)      8*\/ -2 + x      /         2/3     -2 + x \                                                     /         2/3     -2 + x \||
  |                               |                                                                       6*|- ----------- + ------------ + ------------|   2*|----------- - ----------- - ------------ + ---------------------|   7*|2*(1 + x)    + ---------|   22*|2*(1 + x)    + ---------|   |----- + ------|*|- ----------- + ------------ + ------------|   3*|- ----------- + ------------ + ------------|   2*|----- + ------|*|2*(1 + x)    + ---------|   4*|----- + ------|*|2*(1 + x)    + ---------|   16*|2*(1 + x)    + ---------||            |  - ----------- + ------------ + ------------   2*|2*(1 + x)    + ---------|                                                   4*|2*(1 + x)    + ---------|||
  |                               |                                                                         |          4/3           2/3     3 _______  |     |        7/3           5/3           4/3    3 _______         2/3|     |               3 _______|      |               3 _______|   \1 + x   -2 + x/ |          4/3           2/3     3 _______  |     |          4/3           2/3     3 _______  |     \1 + x   -2 + x/ |               3 _______|     \1 + x   -2 + x/ |               3 _______|      |               3 _______||            |            4/3           2/3     3 _______       |               3 _______|                                                     |               3 _______|||
  |           2/3     -2 + x      |/         2/3     -2 + x \ /   5           14             8        \     \   (1 + x)      (-2 + x)        \/ 1 + x   /     \ (1 + x)      (-2 + x)       (1 + x)       \/ 1 + x *(-2 + x)   /     \               \/ 1 + x /      \               \/ 1 + x /                    \   (1 + x)      (-2 + x)        \/ 1 + x   /     \   (1 + x)      (-2 + x)        \/ 1 + x   /                      \               \/ 1 + x /                      \               \/ 1 + x /      \               \/ 1 + x /|            |     (1 + x)      (-2 + x)        \/ 1 + x        \               \/ 1 + x /     /  1       2   \ /         2/3     -2 + x \     \               \/ 1 + x /||
  |  2*(1 + x)    + ---------   x*||2*(1 + x)    + ---------|*|-------- + --------- + ----------------| - ----------------------------------------------- + -------------------------------------------------------------------- + ---------------------------- + ----------------------------- - -------------------------------------------------------------- - ----------------------------------------------- + --------------------------------------------- + --------------------------------------------- + -----------------------------|   (-1 + x)*|- ------------------------------------------- + ---------------------------- + 2*|----- + ------|*|2*(1 + x)    + ---------| + ----------------------------||
  |                 3 _______     ||               3 _______| |       2           2   (1 + x)*(-2 + x)|                             4/3                                                  3 ________                                                 2                               2                                       3 ________                                                    3 ________                                     1 + x                                           -2 + x                             (1 + x)*(-2 + x)      |            |                   3 ________                              1 + x                 \1 + x   -2 + x/ |               3 _______|              -2 + x           ||
  |                 \/ 1 + x      \\               \/ 1 + x / \(1 + x)    (-2 + x)                    /                     (-2 + x)                                                     \/ -2 + x                                           (1 + x)                        (-2 + x)                                        \/ -2 + x                                             (1 + x)*\/ -2 + x                                                                                                                                               /            \                   \/ -2 + x                                                                      \               \/ 1 + x /                               /|
4*|- ------------------------ - ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------------------------------------------------------|
  \           -2 + x                                                                                                                                                                                                                                                                     27                                                                                                                                                                                                                                                                                                                                         3*(-2 + x)                                                                              /
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                           4/3         4/3                                                                                                                                                                                                                                                                                                                                                   
                                                                                                                                                                                                                                                                                                                                                    (1 + x)   *(-2 + x)                                                                                                                                                                                                                                                                                                                                                      
4(x((x2x+13+2(x+1)23)(5(x+1)2+8(x2)(x+1)+14(x2)2)+2(x2x+13+2(x+1)23)(1x+1+2x2)x+1+7(x2x+13+2(x+1)23)(x+1)2+4(x2x+13+2(x+1)23)(1x+1+2x2)x2+16(x2x+13+2(x+1)23)(x2)(x+1)+22(x2x+13+2(x+1)23)(x2)2(1x+1+2x2)((x2)43(x+1)43+8x23x+13+2(x+1)23(x2)23)x23+2((x2)43(x+1)733x23(x+1)43+3(x2)23x+13(x+1)23(x2)53)x233((x2)43(x+1)43+8x23x+13+2(x+1)23(x2)23)x23(x+1)6((x2)43(x+1)43+8x23x+13+2(x+1)23(x2)23)(x2)43)27+(x1)(2(x2x+13+2(x+1)23)(1x+1+2x2)+2(x2x+13+2(x+1)23)x+1+4(x2x+13+2(x+1)23)x2(x2)43(x+1)43+8x23x+13+2(x+1)23(x2)23x23)3(x2)x2x+13+2(x+1)23x2)(x2)43(x+1)43\frac{4 \left(- \frac{x \left(\left(\frac{x - 2}{\sqrt[3]{x + 1}} + 2 \left(x + 1\right)^{\frac{2}{3}}\right) \left(\frac{5}{\left(x + 1\right)^{2}} + \frac{8}{\left(x - 2\right) \left(x + 1\right)} + \frac{14}{\left(x - 2\right)^{2}}\right) + \frac{2 \left(\frac{x - 2}{\sqrt[3]{x + 1}} + 2 \left(x + 1\right)^{\frac{2}{3}}\right) \left(\frac{1}{x + 1} + \frac{2}{x - 2}\right)}{x + 1} + \frac{7 \left(\frac{x - 2}{\sqrt[3]{x + 1}} + 2 \left(x + 1\right)^{\frac{2}{3}}\right)}{\left(x + 1\right)^{2}} + \frac{4 \left(\frac{x - 2}{\sqrt[3]{x + 1}} + 2 \left(x + 1\right)^{\frac{2}{3}}\right) \left(\frac{1}{x + 1} + \frac{2}{x - 2}\right)}{x - 2} + \frac{16 \left(\frac{x - 2}{\sqrt[3]{x + 1}} + 2 \left(x + 1\right)^{\frac{2}{3}}\right)}{\left(x - 2\right) \left(x + 1\right)} + \frac{22 \left(\frac{x - 2}{\sqrt[3]{x + 1}} + 2 \left(x + 1\right)^{\frac{2}{3}}\right)}{\left(x - 2\right)^{2}} - \frac{\left(\frac{1}{x + 1} + \frac{2}{x - 2}\right) \left(- \frac{\left(x - 2\right)^{\frac{4}{3}}}{\left(x + 1\right)^{\frac{4}{3}}} + \frac{8 \sqrt[3]{x - 2}}{\sqrt[3]{x + 1}} + \frac{2 \left(x + 1\right)^{\frac{2}{3}}}{\left(x - 2\right)^{\frac{2}{3}}}\right)}{\sqrt[3]{x - 2}} + \frac{2 \left(\frac{\left(x - 2\right)^{\frac{4}{3}}}{\left(x + 1\right)^{\frac{7}{3}}} - \frac{3 \sqrt[3]{x - 2}}{\left(x + 1\right)^{\frac{4}{3}}} + \frac{3}{\left(x - 2\right)^{\frac{2}{3}} \sqrt[3]{x + 1}} - \frac{\left(x + 1\right)^{\frac{2}{3}}}{\left(x - 2\right)^{\frac{5}{3}}}\right)}{\sqrt[3]{x - 2}} - \frac{3 \left(- \frac{\left(x - 2\right)^{\frac{4}{3}}}{\left(x + 1\right)^{\frac{4}{3}}} + \frac{8 \sqrt[3]{x - 2}}{\sqrt[3]{x + 1}} + \frac{2 \left(x + 1\right)^{\frac{2}{3}}}{\left(x - 2\right)^{\frac{2}{3}}}\right)}{\sqrt[3]{x - 2} \left(x + 1\right)} - \frac{6 \left(- \frac{\left(x - 2\right)^{\frac{4}{3}}}{\left(x + 1\right)^{\frac{4}{3}}} + \frac{8 \sqrt[3]{x - 2}}{\sqrt[3]{x + 1}} + \frac{2 \left(x + 1\right)^{\frac{2}{3}}}{\left(x - 2\right)^{\frac{2}{3}}}\right)}{\left(x - 2\right)^{\frac{4}{3}}}\right)}{27} + \frac{\left(x - 1\right) \left(2 \left(\frac{x - 2}{\sqrt[3]{x + 1}} + 2 \left(x + 1\right)^{\frac{2}{3}}\right) \left(\frac{1}{x + 1} + \frac{2}{x - 2}\right) + \frac{2 \left(\frac{x - 2}{\sqrt[3]{x + 1}} + 2 \left(x + 1\right)^{\frac{2}{3}}\right)}{x + 1} + \frac{4 \left(\frac{x - 2}{\sqrt[3]{x + 1}} + 2 \left(x + 1\right)^{\frac{2}{3}}\right)}{x - 2} - \frac{- \frac{\left(x - 2\right)^{\frac{4}{3}}}{\left(x + 1\right)^{\frac{4}{3}}} + \frac{8 \sqrt[3]{x - 2}}{\sqrt[3]{x + 1}} + \frac{2 \left(x + 1\right)^{\frac{2}{3}}}{\left(x - 2\right)^{\frac{2}{3}}}}{\sqrt[3]{x - 2}}\right)}{3 \left(x - 2\right)} - \frac{\frac{x - 2}{\sqrt[3]{x + 1}} + 2 \left(x + 1\right)^{\frac{2}{3}}}{x - 2}\right)}{\left(x - 2\right)^{\frac{4}{3}} \left(x + 1\right)^{\frac{4}{3}}}