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

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

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

Solución

Ha introducido [src]

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

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

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

Solución detallada

Sustituimos $u = \frac{x \left(x^{2} + 1\right)}{\left(x^{2} - 1\right)^{3}}$ .
Según el principio, aplicamos: $\sqrt{u}$ tenemos $\frac{1}{2 \sqrt{u}}$
Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \frac{x \left(x^{2} + 1\right)}{\left(x^{2} - 1\right)^{3}}$ :
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 $g{\left(x \right)} = \left(x^{2} - 1\right)^{3}$ .
  
  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: $u^{3}$ tenemos $3 u^{2}$
  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:
    
    $6 x \left(x^{2} - 1\right)^{2}$
  Ahora aplicamos la regla de la derivada de una divesión:
  
  $\frac{- 6 x^{2} \left(x^{2} - 1\right)^{2} \left(x^{2} + 1\right) + \left(x^{2} - 1\right)^{3} \left(3 x^{2} + 1\right)}{\left(x^{2} - 1\right)^{6}}$
Como resultado de la secuencia de reglas:

$\frac{- 6 x^{2} \left(x^{2} - 1\right)^{2} \left(x^{2} + 1\right) + \left(x^{2} - 1\right)^{3} \left(3 x^{2} + 1\right)}{2 \sqrt{\frac{x}{\left(x^{2} - 1\right)^{3}}} \left(x^{2} - 1\right)^{6} \sqrt{x^{2} + 1}}$
Simplificamos:

$\frac{- 6 x^{2} \left(x^{2} + 1\right) + \left(x^{2} - 1\right) \left(3 x^{2} + 1\right)}{2 \sqrt{\frac{x}{\left(x^{2} - 1\right)^{3}}} \left(x^{2} - 1\right)^{4} \sqrt{x^{2} + 1}}$

Respuesta:

$\frac{- 6 x^{2} \left(x^{2} + 1\right) + \left(x^{2} - 1\right) \left(3 x^{2} + 1\right)}{2 \sqrt{\frac{x}{\left(x^{2} - 1\right)^{3}}} \left(x^{2} - 1\right)^{4} \sqrt{x^{2} + 1}}$

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

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

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

Simplificar

Segunda derivada [src]

                   /                   2 /     2\                  2 /     2\                                           /               2 /     2\\                    /              2 /     2\\ /               2 /     2\\\
                   |            2   6*x *\1 + x /           2   6*x *\1 + x /                                           |        2   6*x *\1 + x /|                    |       2   6*x *\1 + x /| |        2   6*x *\1 + x /||
                   |    -1 - 3*x  + -------------   -1 - 3*x  + -------------                                         3*|-1 - 3*x  + -------------|                    |1 + 3*x  - -------------|*|-1 - 3*x  + -------------||
      ____________ |                         2                           2           2      /     2\     /       2\     |                     2   |       2 /     2\   |                    2   | |                     2   ||
     /     x       |                   -1 + x                      -1 + x         6*x     6*\1 + x /   3*\1 + 3*x /     \               -1 + x    /   24*x *\1 + x /   \              -1 + x    / \               -1 + x    /|
    /  ---------- *|3 + ------------------------- + ------------------------- - ------- - ---------- - ------------ - ----------------------------- + -------------- - ------------------------------------------------------|
   /            3  |                   2                          2                   2          2             2                       2                         2                            2 /     2\                     |
  /    /      2\   |              1 + x                        2*x              -1 + x     -1 + x        -1 + x                  -1 + x                 /      2\                          4*x *\1 + x /                     |
\/     \-1 + x /   \                                                                                                                                    \-1 + x /                                                            /
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                            ________                                                                                                          
                                                                                                           /      2                                                                                                           
                                                                                                         \/  1 + x

$$\frac{\sqrt{\frac{x}{\left(x^{2} - 1\right)^{3}}} \left(- \frac{6 x^{2}}{x^{2} - 1} + \frac{24 x^{2} \left(x^{2} + 1\right)}{\left(x^{2} - 1\right)^{2}} + 3 + \frac{- 3 x^{2} + \frac{6 x^{2} \left(x^{2} + 1\right)}{x^{2} - 1} - 1}{x^{2} + 1} - \frac{6 \left(x^{2} + 1\right)}{x^{2} - 1} - \frac{3 \left(3 x^{2} + 1\right)}{x^{2} - 1} - \frac{3 \left(- 3 x^{2} + \frac{6 x^{2} \left(x^{2} + 1\right)}{x^{2} - 1} - 1\right)}{x^{2} - 1} + \frac{- 3 x^{2} + \frac{6 x^{2} \left(x^{2} + 1\right)}{x^{2} - 1} - 1}{2 x^{2}} - \frac{\left(- 3 x^{2} + \frac{6 x^{2} \left(x^{2} + 1\right)}{x^{2} - 1} - 1\right) \left(3 x^{2} - \frac{6 x^{2} \left(x^{2} + 1\right)}{x^{2} - 1} + 1\right)}{4 x^{2} \left(x^{2} + 1\right)}\right)}{\sqrt{x^{2} + 1}}$$

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\     |    1 + 3*x     22*x     2*\1 + x /     32*x       80*x *\1 + x /   8*x *\1 + 3*x /   40*x *\1 + x /|     |     1 + 3*x      2*x     2*\1 + x /   8*x *\1 + x /|        |     1 + 3*x      2*x     2*\1 + x /   8*x *\1 + x /|        /               2 /     2\\       /               2 /     2\\     /               2 /     2\\       |     1 + 3*x      2*x     2*\1 + x /   8*x *\1 + x /|                  2 /     2\   /          2 \ /               2 /     2\\     |       6*x  | |     1 + 3*x      2*x     2*\1 + x /   8*x *\1 + x /|   /              2 /     2\\ /               2 /     2\\       /               2 /     2\\     |       2   6*x *\1 + x /| |     1 + 3*x      2*x     2*\1 + x /   8*x *\1 + x /|   /          2 \ /               2 /     2\\     |        2   6*x *\1 + x /| |     3*\1 + x /     4*x     8*x *\1 + x /|     /          2 \ /               2 /     2\\     /              2 /     2\\ /               2 /     2\\     /              2 /     2\\ /               2 /     2\\   /          2 \ /              2 /     2\\ /               2 /     2\\|
                   |          2   6*x *\1 + x /   3*|1 - -------- - ------- - ---------- + ---------- - -------------- + --------------- + --------------|   6*|-1 + -------- + ------- + ---------- - -------------|   36*x*|-1 + -------- + ------- + ---------- - -------------|        |        2   6*x *\1 + x /|       |        2   6*x *\1 + x /|     |        2   6*x *\1 + x /|   9*x*|-1 + -------- + ------- + ---------- - -------------|           2   6*x *\1 + x /   |       6*x  | |        2   6*x *\1 + x /|   3*|-1 + -------|*|-1 + -------- + ------- + ---------- - -------------|   |       2   6*x *\1 + x /| |        2   6*x *\1 + x /|       |        2   6*x *\1 + x /|   3*|1 + 3*x  - -------------|*|-1 + -------- + ------- + ---------- - -------------|   |       6*x  | |        2   6*x *\1 + x /|   3*|-1 - 3*x  + -------------|*|-1 + ---------- + ------- - -------------|     |       6*x  | |        2   6*x *\1 + x /|     |       2   6*x *\1 + x /| |        2   6*x *\1 + x /|     |       2   6*x *\1 + x /| |        2   6*x *\1 + x /|   |       6*x  | |       2   6*x *\1 + x /| |        2   6*x *\1 + x /||
                   |  -1 - 3*x  + -------------     |          2          2          2              2              3                 2                2  |     |           2          2          2                2 |        |           2          2          2                2 |   12*x*|-1 - 3*x  + -------------|   3*x*|-1 - 3*x  + -------------|   3*|-1 - 3*x  + -------------|       |           2          2          2                2 |   -1 - 3*x  + -------------   |-1 + -------|*|-1 - 3*x  + -------------|     |           2| |           2          2          2                2 |   |1 + 3*x  - -------------|*|-1 - 3*x  + -------------|   9*x*|-1 - 3*x  + -------------|     |                    2   | |           2          2          2                2 |   |-1 + -------|*|-1 - 3*x  + -------------|     |                     2   | |            2           2              2 |   3*|-1 + -------|*|-1 - 3*x  + -------------|   3*|1 + 3*x  - -------------|*|-1 - 3*x  + -------------|   3*|1 + 3*x  - -------------|*|-1 - 3*x  + -------------|   |-1 + -------|*|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   |       |     -1 + x     -1 + x     -1 + x        /      2\  |                        2      |           2| |                     2   |     \     -1 + x / |     -1 + x     -1 + x     -1 + x        /      2\  |   |                    2   | |                     2   |       |                     2   |     \              -1 + x    / |     -1 + x     -1 + x     -1 + x        /      2\  |   |           2| |                     2   |     \               -1 + x    / |      -1 + x      -1 + x      /      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    /|
    /  ---------- *|- ------------------------- + -------------------------------------------------------------------------------------------------------- + -------------------------------------------------------- - ----------------------------------------------------------- - -------------------------------- - ------------------------------- + ----------------------------- + ---------------------------------------------------------- - ------------------------- - ------------------------------------------ + ----------------------------------------------------------------------- + ------------------------------------------------------ + ------------------------------- - ----------------------------------------------------------------------------------- - ------------------------------------------ + ------------------------------------------------------------------------- + -------------------------------------------- + -------------------------------------------------------- - -------------------------------------------------------- + ---------------------------------------------------------------------|
   /            3  |               3                                                                 x                                                                                  x                                                               2                                                 2                                 2                         /      2\                                           2                                        /     2\                               3                                                        2*x                                                            3 /     2\                               /     2\ /      2\                                                 /     2\                                                         /     2\                                                     /     2\                                                    /      2\                                                    2                                             /     2\ /      2\                                                    3 /     2\                            |
  /    /      2\   |              x                                                                                                                                                                                                               -1 + x                                         /      2\                          /     2\                        x*\-1 + x /                                      1 + x                                     2*x*\1 + x /                            4*x                                                                                                                     2*x *\1 + x /                               \1 + x /*\-1 + x /                                             2*x*\1 + x /                                                     2*x*\1 + x /                                                 2*x*\1 + x /                                                2*x*\-1 + x /                                            /     2\                                            x*\1 + x /*\-1 + x /                                                 8*x *\1 + x /                            |
\/     \-1 + x /   \                                                                                                                                                                                                                                                                             \-1 + x /                          \1 + x /                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             4*x*\1 + x /                                                                                                                                                          /
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                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            /      2                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          \/  1 + x

$$\frac{\sqrt{\frac{x}{\left(x^{2} - 1\right)^{3}}} \left(\frac{9 x \left(\frac{2 x^{2}}{x^{2} - 1} - \frac{8 x^{2} \left(x^{2} + 1\right)}{\left(x^{2} - 1\right)^{2}} - 1 + \frac{2 \left(x^{2} + 1\right)}{x^{2} - 1} + \frac{3 x^{2} + 1}{x^{2} - 1}\right)}{x^{2} + 1} - \frac{3 x \left(- 3 x^{2} + \frac{6 x^{2} \left(x^{2} + 1\right)}{x^{2} - 1} - 1\right)}{\left(x^{2} + 1\right)^{2}} - \frac{36 x \left(\frac{2 x^{2}}{x^{2} - 1} - \frac{8 x^{2} \left(x^{2} + 1\right)}{\left(x^{2} - 1\right)^{2}} - 1 + \frac{2 \left(x^{2} + 1\right)}{x^{2} - 1} + \frac{3 x^{2} + 1}{x^{2} - 1}\right)}{x^{2} - 1} + \frac{9 x \left(- 3 x^{2} + \frac{6 x^{2} \left(x^{2} + 1\right)}{x^{2} - 1} - 1\right)}{\left(x^{2} - 1\right) \left(x^{2} + 1\right)} - \frac{12 x \left(- 3 x^{2} + \frac{6 x^{2} \left(x^{2} + 1\right)}{x^{2} - 1} - 1\right)}{\left(x^{2} - 1\right)^{2}} + \frac{3 \left(\frac{6 x^{2}}{x^{2} - 1} - 1\right) \left(\frac{2 x^{2}}{x^{2} - 1} - \frac{8 x^{2} \left(x^{2} + 1\right)}{\left(x^{2} - 1\right)^{2}} - 1 + \frac{2 \left(x^{2} + 1\right)}{x^{2} - 1} + \frac{3 x^{2} + 1}{x^{2} - 1}\right)}{2 x} + \frac{6 \left(\frac{2 x^{2}}{x^{2} - 1} - \frac{8 x^{2} \left(x^{2} + 1\right)}{\left(x^{2} - 1\right)^{2}} - 1 + \frac{2 \left(x^{2} + 1\right)}{x^{2} - 1} + \frac{3 x^{2} + 1}{x^{2} - 1}\right)}{x} + \frac{3 \left(\frac{32 x^{4}}{\left(x^{2} - 1\right)^{2}} - \frac{80 x^{4} \left(x^{2} + 1\right)}{\left(x^{2} - 1\right)^{3}} - \frac{22 x^{2}}{x^{2} - 1} + \frac{40 x^{2} \left(x^{2} + 1\right)}{\left(x^{2} - 1\right)^{2}} + \frac{8 x^{2} \left(3 x^{2} + 1\right)}{\left(x^{2} - 1\right)^{2}} + 1 - \frac{2 \left(x^{2} + 1\right)}{x^{2} - 1} - \frac{3 x^{2} + 1}{x^{2} - 1}\right)}{x} - \frac{\left(\frac{6 x^{2}}{x^{2} - 1} - 1\right) \left(- 3 x^{2} + \frac{6 x^{2} \left(x^{2} + 1\right)}{x^{2} - 1} - 1\right)}{2 x \left(x^{2} + 1\right)} + \frac{3 \left(- 3 x^{2} + \frac{6 x^{2} \left(x^{2} + 1\right)}{x^{2} - 1} - 1\right) \left(\frac{4 x^{2}}{x^{2} - 1} - \frac{8 x^{2} \left(x^{2} + 1\right)}{\left(x^{2} - 1\right)^{2}} - 1 + \frac{3 \left(x^{2} + 1\right)}{x^{2} - 1}\right)}{2 x \left(x^{2} + 1\right)} - \frac{- 3 x^{2} + \frac{6 x^{2} \left(x^{2} + 1\right)}{x^{2} - 1} - 1}{2 x \left(x^{2} + 1\right)} - \frac{3 \left(3 x^{2} - \frac{6 x^{2} \left(x^{2} + 1\right)}{x^{2} - 1} + 1\right) \left(\frac{2 x^{2}}{x^{2} - 1} - \frac{8 x^{2} \left(x^{2} + 1\right)}{\left(x^{2} - 1\right)^{2}} - 1 + \frac{2 \left(x^{2} + 1\right)}{x^{2} - 1} + \frac{3 x^{2} + 1}{x^{2} - 1}\right)}{2 x \left(x^{2} + 1\right)} + \frac{3 \left(- 3 x^{2} + \frac{6 x^{2} \left(x^{2} + 1\right)}{x^{2} - 1} - 1\right) \left(3 x^{2} - \frac{6 x^{2} \left(x^{2} + 1\right)}{x^{2} - 1} + 1\right)}{4 x \left(x^{2} + 1\right)^{2}} + \frac{3 \left(\frac{6 x^{2}}{x^{2} - 1} - 1\right) \left(- 3 x^{2} + \frac{6 x^{2} \left(x^{2} + 1\right)}{x^{2} - 1} - 1\right)}{2 x \left(x^{2} - 1\right)} + \frac{3 \left(- 3 x^{2} + \frac{6 x^{2} \left(x^{2} + 1\right)}{x^{2} - 1} - 1\right)}{x \left(x^{2} - 1\right)} - \frac{3 \left(- 3 x^{2} + \frac{6 x^{2} \left(x^{2} + 1\right)}{x^{2} - 1} - 1\right) \left(3 x^{2} - \frac{6 x^{2} \left(x^{2} + 1\right)}{x^{2} - 1} + 1\right)}{x \left(x^{2} - 1\right) \left(x^{2} + 1\right)} - \frac{\left(\frac{6 x^{2}}{x^{2} - 1} - 1\right) \left(- 3 x^{2} + \frac{6 x^{2} \left(x^{2} + 1\right)}{x^{2} - 1} - 1\right)}{4 x^{3}} - \frac{- 3 x^{2} + \frac{6 x^{2} \left(x^{2} + 1\right)}{x^{2} - 1} - 1}{x^{3}} + \frac{\left(\frac{6 x^{2}}{x^{2} - 1} - 1\right) \left(- 3 x^{2} + \frac{6 x^{2} \left(x^{2} + 1\right)}{x^{2} - 1} - 1\right) \left(3 x^{2} - \frac{6 x^{2} \left(x^{2} + 1\right)}{x^{2} - 1} + 1\right)}{8 x^{3} \left(x^{2} + 1\right)} + \frac{\left(- 3 x^{2} + \frac{6 x^{2} \left(x^{2} + 1\right)}{x^{2} - 1} - 1\right) \left(3 x^{2} - \frac{6 x^{2} \left(x^{2} + 1\right)}{x^{2} - 1} + 1\right)}{2 x^{3} \left(x^{2} + 1\right)}\right)}{\sqrt{x^{2} + 1}}$$

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Gráfico

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Expresiones semejantes

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

Gráfico:

Definida a trozos:

Solución