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Expresiones semejantes
(x*(x^2+1)/(x^2+1)^2)^(1/3)
(x*(x^2-1)/(x^2-1)^2)^(1/3)

Derivada de la función/ (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)

Solución

Ha introducido [src]

       ____________
      /   / 2    \ 
     /  x*\x  + 1/ 
    /   ---------- 
   /            2  
3 /     / 2    \   
\/      \x  - 1/

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

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

Solución detallada

Sustituimos $u = \frac{x \left(x^{2} + 1\right)}{\left(x^{2} - 1\right)^{2}}$ .
Según el principio, aplicamos: $\sqrt[3]{u}$ tenemos $\frac{1}{3 u^{\frac{2}{3}}}$
Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \frac{x \left(x^{2} + 1\right)}{\left(x^{2} - 1\right)^{2}}$ :
1. Se aplica la regla de la derivada parcial:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = x \left(x^{2} + 1\right)$ y $g{\left(x \right)} = \left(x^{2} - 1\right)^{2}$ .
  
  Para calcular $\frac{d}{d x} f{\left(x \right)}$ :
  1. Se aplica la regla de la derivada de una multiplicación:
    
    $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
    
    $f{\left(x \right)} = x$ ; calculamos $\frac{d}{d x} f{\left(x \right)}$ :
    1. Según el principio, aplicamos: $x$ tenemos $1$
    $g{\left(x \right)} = x^{2} + 1$ ; calculamos $\frac{d}{d x} g{\left(x \right)}$ :
    1. diferenciamos $x^{2} + 1$ miembro por miembro:
      1. La derivada de una constante $1$ es igual a cero.
      2. Según el principio, aplicamos: $x^{2}$ tenemos $2 x$
      Como resultado de: $2 x$
    Como resultado de: $3 x^{2} + 1$
  Para calcular $\frac{d}{d x} g{\left(x \right)}$ :
  1. Sustituimos $u = x^{2} - 1$ .
  2. Según el principio, aplicamos: $u^{2}$ tenemos $2 u$
  3. Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \left(x^{2} - 1\right)$ :
    1. diferenciamos $x^{2} - 1$ miembro por miembro:
      1. La derivada de una constante $-1$ es igual a cero.
      2. Según el principio, aplicamos: $x^{2}$ tenemos $2 x$
      Como resultado de: $2 x$
    Como resultado de la secuencia de reglas:
    
    $2 x \left(2 x^{2} - 2\right)$
  Ahora aplicamos la regla de la derivada de una divesión:
  
  $\frac{- 2 x^{2} \left(x^{2} + 1\right) \left(2 x^{2} - 2\right) + \left(x^{2} - 1\right)^{2} \left(3 x^{2} + 1\right)}{\left(x^{2} - 1\right)^{4}}$
Como resultado de la secuencia de reglas:

$\frac{- 2 x^{2} \left(x^{2} + 1\right) \left(2 x^{2} - 2\right) + \left(x^{2} - 1\right)^{2} \left(3 x^{2} + 1\right)}{3 \left(\frac{x}{\left(x^{2} - 1\right)^{2}}\right)^{\frac{2}{3}} \left(x^{2} - 1\right)^{4} \left(x^{2} + 1\right)^{\frac{2}{3}}}$
Simplificamos:

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

Respuesta:

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

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

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/

$$\frac{\sqrt[3]{\frac{x \left(x^{2} + 1\right)}{\left(x^{2} - 1\right)^{2}}} \left(x^{2} - 1\right)^{2} \left(- \frac{4 x^{2} \left(x^{2} + 1\right)}{3 \left(x^{2} - 1\right)^{3}} + \frac{3 x^{2} + 1}{3 \left(x^{2} - 1\right)^{2}}\right)}{x \left(x^{2} + 1\right)}$$

Simplificar

Segunda derivada [src]

                   /                                                 /               2 /     2\\                  2 /     2\     /               2 /     2\\                   /              2 /     2\\ /               2 /     2\\\
                   |                                                 |        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  + -------------||
      ____________ |           2         /     2\     /       2\     |                     2   |                        2        |                     2   |      2 /     2\   |                    2   | |                     2   ||
     /     x       |        8*x        8*\1 + x /   4*\1 + 3*x /     \               -1 + x    /                  -1 + x         \               -1 + x    /   8*x *\1 + x /   \              -1 + x    / \               -1 + x    /|
    /  ---------- *|2 - ----------- - ----------- - ------------ - ----------------------------- + ------------------------- + ----------------------------- + ------------- - ------------------------------------------------------|
   /            2  |      /      2\     /      2\     /      2\               /      2\                          2                         /     2\                       2                           2 /     2\                     |
3 /    /      2\   |    3*\-1 + x /   3*\-1 + x /   3*\-1 + x /             3*\-1 + x /                       3*x                        3*\1 + x /              /      2\                         9*x *\1 + x /                     |
\/     \-1 + x /   \                                                                                                                                             \-1 + x /                                                           /
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                                                                                                                     2/3                                                                                                              
                                                                                                             /     2\                                                                                                                 
                                                                                                             \1 + x /

$$\frac{\sqrt[3]{\frac{x}{\left(x^{2} - 1\right)^{2}}} \left(- \frac{8 x^{2}}{3 \left(x^{2} - 1\right)} + \frac{8 x^{2} \left(x^{2} + 1\right)}{\left(x^{2} - 1\right)^{2}} + 2 + \frac{2 \left(- 3 x^{2} + \frac{4 x^{2} \left(x^{2} + 1\right)}{x^{2} - 1} - 1\right)}{3 \left(x^{2} + 1\right)} - \frac{8 \left(x^{2} + 1\right)}{3 \left(x^{2} - 1\right)} - \frac{4 \left(3 x^{2} + 1\right)}{3 \left(x^{2} - 1\right)} - \frac{4 \left(- 3 x^{2} + \frac{4 x^{2} \left(x^{2} + 1\right)}{x^{2} - 1} - 1\right)}{3 \left(x^{2} - 1\right)} + \frac{- 3 x^{2} + \frac{4 x^{2} \left(x^{2} + 1\right)}{x^{2} - 1} - 1}{3 x^{2}} - \frac{\left(- 3 x^{2} + \frac{4 x^{2} \left(x^{2} + 1\right)}{x^{2} - 1} - 1\right) \left(3 x^{2} - \frac{4 x^{2} \left(x^{2} + 1\right)}{x^{2} - 1} + 1\right)}{9 x^{2} \left(x^{2} + 1\right)}\right)}{\left(x^{2} + 1\right)^{\frac{2}{3}}}$$

Simplificar

Tercera derivada [src]

                   /                                      /         2      /     2\     /       2\         4          4 /     2\       2 /       2\       2 /     2\\      /       /       2\        2      /     2\       2 /     2\\         /       /       2\        2      /     2\       2 /     2\\                                                                                                                                                           /          2 \ /       /       2\        2      /     2\       2 /     2\\                                         /       /       2\        2      /     2\       2 /     2\\                                                    /              2 /     2\\ /       /       2\        2      /     2\       2 /     2\\     /               2 /     2\\ /       /     2\        2        2 /     2\\                                                                                                                                                                                                                                                                                                                                               \
                   |     /               2 /     2\\      |     44*x     4*\1 + x /   2*\1 + 3*x /     48*x       96*x *\1 + x /   12*x *\1 + 3*x /   60*x *\1 + x /|      |     2*\1 + 3*x /     4*x     4*\1 + x /   12*x *\1 + x /|         |     2*\1 + 3*x /     4*x     4*\1 + x /   12*x *\1 + x /|        /               2 /     2\\        /               2 /     2\\      /               2 /     2\\     /          2 \ /               2 /     2\\     |       4*x  | |     2*\1 + 3*x /     4*x     4*\1 + x /   12*x *\1 + x /|      /               2 /     2\\        |     2*\1 + 3*x /     4*x     4*\1 + x /   12*x *\1 + x /|     /          2 \ /               2 /     2\\     |       2   4*x *\1 + x /| |     2*\1 + 3*x /     4*x     4*\1 + x /   12*x *\1 + x /|     |        2   4*x *\1 + x /| |     6*\1 + x /     8*x     12*x *\1 + x /|     /              2 /     2\\ /               2 /     2\\      /              2 /     2\\ /               2 /     2\\      /          2 \ /               2 /     2\\         /               2 /     2\\   /          2 \ /              2 /     2\\ /               2 /     2\\      /              2 /     2\\ /               2 /     2\\|
                   |     |        2   4*x *\1 + x /|   18*|3 - ------- - ---------- - ------------ + ---------- - -------------- + ---------------- + --------------|   36*|-3 + ------------ + ------- + ---------- - --------------|   144*x*|-3 + ------------ + ------- + ---------- - --------------|        |        2   4*x *\1 + x /|        |        2   4*x *\1 + x /|      |        2   4*x *\1 + x /|     |       4*x  | |        2   4*x *\1 + x /|   6*|-1 + -------|*|-3 + ------------ + ------- + ---------- - --------------|      |        2   4*x *\1 + x /|   60*x*|-3 + ------------ + ------- + ---------- - --------------|     |       4*x  | |        2   4*x *\1 + x /|   6*|1 + 3*x  - -------------|*|-3 + ------------ + ------- + ---------- - --------------|   6*|-1 - 3*x  + -------------|*|-3 + ---------- + ------- - --------------|     |       2   4*x *\1 + x /| |        2   4*x *\1 + x /|      |       2   4*x *\1 + x /| |        2   4*x *\1 + x /|      |       4*x  | |        2   4*x *\1 + x /|         |        2   4*x *\1 + x /|   |       4*x  | |       2   4*x *\1 + x /| |        2   4*x *\1 + x /|      |       2   4*x *\1 + x /| |        2   4*x *\1 + x /||
                   |  18*|-1 - 3*x  + -------------|      |          2          2             2               2              3                 2                 2  |      |             2            2          2                2  |         |             2            2          2                2  |   72*x*|-1 - 3*x  + -------------|   60*x*|-1 - 3*x  + -------------|   12*|-1 - 3*x  + -------------|   3*|-1 + -------|*|-1 - 3*x  + -------------|     |           2| |             2            2          2                2  |   36*|-1 - 3*x  + -------------|        |             2            2          2                2  |   6*|-1 + -------|*|-1 - 3*x  + -------------|     |                    2   | |             2            2          2                2  |     |                     2   | |            2           2              2  |   6*|1 + 3*x  - -------------|*|-1 - 3*x  + -------------|   10*|1 + 3*x  - -------------|*|-1 - 3*x  + -------------|   12*|-1 + -------|*|-1 - 3*x  + -------------|   120*x*|-1 - 3*x  + -------------|   |-1 + -------|*|1 + 3*x  - -------------|*|-1 - 3*x  + -------------|   24*|1 + 3*x  - -------------|*|-1 - 3*x  + -------------||
      ____________ |     |                     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    /|
    /  ---------- *|- ------------------------------ + -------------------------------------------------------------------------------------------------------------- + -------------------------------------------------------------- - ----------------------------------------------------------------- - -------------------------------- - -------------------------------- - ------------------------------ - -------------------------------------------- + ---------------------------------------------------------------------------- + ------------------------------ + ---------------------------------------------------------------- - -------------------------------------------- - ---------------------------------------------------------------------------------------- + -------------------------------------------------------------------------- + -------------------------------------------------------- + --------------------------------------------------------- + --------------------------------------------- + --------------------------------- + --------------------------------------------------------------------- - ---------------------------------------------------------|
   /            2  |                 3                                                                       x                                                                                        x                                                                     2                                                    2                                 2                           /     2\                                   3                                                             x                                                    /      2\                                               2                                                   /     2\                                                             /     2\                                                                            /     2\                                                          3 /     2\                                                           2                                             /      2\                            /     2\ /      2\                                        3 /     2\                                                     /     2\ /      2\                  |
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 /

$$\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

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Expresiones idénticas

Expresiones semejantes

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

Gráfico:

Definida a trozos:

Solución