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

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

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

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src]
      ___________
     /       ___ 
    /  x + \/ x  
   /   --------- 
  /          ___ 
\/     x - \/ x  
$$\sqrt{\frac{\sqrt{x} + x}{- \sqrt{x} + x}}$$
sqrt((x + sqrt(x))/(x - sqrt(x)))
Solución detallada
  1. Sustituimos .

  2. Según el principio, aplicamos: tenemos

  3. Luego se aplica una cadena de reglas. Multiplicamos por :

    1. Se aplica la regla de la derivada parcial:

      y .

      Para calcular :

      1. diferenciamos miembro por miembro:

        1. Según el principio, aplicamos: tenemos

        2. Según el principio, aplicamos: tenemos

        Como resultado de:

      Para calcular :

      1. diferenciamos miembro por miembro:

        1. Según el principio, aplicamos: tenemos

        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: tenemos

          Entonces, como resultado:

        Como resultado de:

      Ahora aplicamos la regla de la derivada de una divesión:

    Como resultado de la secuencia de reglas:

  4. Simplificamos:


Respuesta:

Gráfica
Primera derivada [src]
                              /        1       /        1   \ /      ___\\
      ___________             | 1 + -------    |-1 + -------|*\x + \/ x /|
     /       ___              |         ___    |         ___|            |
    /  x + \/ x   /      ___\ |     2*\/ x     \     2*\/ x /            |
   /   --------- *\x - \/ x /*|------------- + --------------------------|
  /          ___              |  /      ___\                      2      |
\/     x - \/ x               |2*\x - \/ x /           /      ___\       |
                              \                      2*\x - \/ x /       /
--------------------------------------------------------------------------
                                      ___                                 
                                x + \/ x                                  
$$\frac{\sqrt{\frac{\sqrt{x} + x}{- \sqrt{x} + x}} \left(- \sqrt{x} + x\right) \left(\frac{\left(-1 + \frac{1}{2 \sqrt{x}}\right) \left(\sqrt{x} + x\right)}{2 \left(- \sqrt{x} + x\right)^{2}} + \frac{1 + \frac{1}{2 \sqrt{x}}}{2 \left(- \sqrt{x} + x\right)}\right)}{\sqrt{x} + x}$$
Segunda derivada [src]
                      /                                              2                                                                                                                                                                                        \
                      |         /            /      1  \ /      ___\\                  /            /      1  \ /      ___\\                 /            /      1  \ /      ___\\                                                                            |
                      |         |            |2 - -----|*\x + \/ x /|                  |            |2 - -----|*\x + \/ x /|                 |            |2 - -----|*\x + \/ x /|                                                                            |
                      |         |            |      ___|            |                  |            |      ___|            |                 |            |      ___|            |                                   2                                        |
                      |         |      1     \    \/ x /            |      /      1  \ |      1     \    \/ x /            |     /      1  \ |      1     \    \/ x /            |                        /      1  \  /      ___\     /      1  \ /      1  \|
      _______________ |         |2 + ----- + -----------------------|    2*|2 + -----|*|2 + ----- + -----------------------|   2*|2 - -----|*|2 + ----- + -----------------------|                      4*|2 - -----| *\x + \/ x /   4*|2 + -----|*|2 - -----||
     /  /      ___\   |         |      ___            ___           |      |      ___| |      ___            ___           |     |      ___| |      ___            ___           |      /      ___\       |      ___|                  |      ___| |      ___||
    /  -\x + \/ x /   |   2     \    \/ x           \/ x  - x       /      \    \/ x / \    \/ x           \/ x  - x       /     \    \/ x / \    \/ x           \/ x  - x       /    2*\x + \/ x /       \    \/ x /                  \    \/ x / \    \/ x /|
   /   ------------- *|- ---- + -------------------------------------- - --------------------------------------------------- - --------------------------------------------------- + ---------------- + -------------------------- + -------------------------|
  /        ___        |   3/2                       ___                                             ___                                               ___                             3/2 /  ___    \                     2                    ___            |
\/       \/ x  - x    |  x                    x + \/ x                                        x + \/ x                                              \/ x  - x                        x   *\\/ x  - x/          /  ___    \                   \/ x  - x        |
                      \                                                                                                                                                                                        \\/ x  - x/                                    /
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                            /      ___\                                                                                                                        
                                                                                                                         16*\x + \/ x /                                                                                                                        
$$\frac{\sqrt{- \frac{\sqrt{x} + x}{\sqrt{x} - x}} \left(\frac{4 \left(2 - \frac{1}{\sqrt{x}}\right)^{2} \left(\sqrt{x} + x\right)}{\left(\sqrt{x} - x\right)^{2}} + \frac{4 \left(2 - \frac{1}{\sqrt{x}}\right) \left(2 + \frac{1}{\sqrt{x}}\right)}{\sqrt{x} - x} - \frac{2 \left(2 - \frac{1}{\sqrt{x}}\right) \left(\frac{\left(2 - \frac{1}{\sqrt{x}}\right) \left(\sqrt{x} + x\right)}{\sqrt{x} - x} + 2 + \frac{1}{\sqrt{x}}\right)}{\sqrt{x} - x} - \frac{2 \left(2 + \frac{1}{\sqrt{x}}\right) \left(\frac{\left(2 - \frac{1}{\sqrt{x}}\right) \left(\sqrt{x} + x\right)}{\sqrt{x} - x} + 2 + \frac{1}{\sqrt{x}}\right)}{\sqrt{x} + x} + \frac{\left(\frac{\left(2 - \frac{1}{\sqrt{x}}\right) \left(\sqrt{x} + x\right)}{\sqrt{x} - x} + 2 + \frac{1}{\sqrt{x}}\right)^{2}}{\sqrt{x} + x} - \frac{2}{x^{\frac{3}{2}}} + \frac{2 \left(\sqrt{x} + x\right)}{x^{\frac{3}{2}} \left(\sqrt{x} - x\right)}\right)}{16 \left(\sqrt{x} + x\right)}$$
Tercera derivada [src]
                      /                                                                                                    /                                         2                                        \                 /                                         2                                        \                                                                                                                                                                                      /                                         2                                        \                                                                                                                                                                                                                                                                                            \
                      |                                            3                                                       |                              /      1  \  /      ___\     /      1  \ /      1  \|                 |                              /      1  \  /      ___\     /      1  \ /      1  \|                                          2                                                                                                     /            /      1  \ /      ___\\ |                              /      1  \  /      ___\     /      1  \ /      1  \|                                                                                                                                                                                2                                                                                                           |
                      |       /            /      1  \ /      ___\\                                                        |                            2*|2 - -----| *\x + \/ x /   2*|2 + -----|*|2 - -----||                 |                            2*|2 - -----| *\x + \/ x /   2*|2 + -----|*|2 - -----||     /            /      1  \ /      ___\\                  /            /      1  \ /      ___\\     /            /      1  \ /      ___\\     |            |2 - -----|*\x + \/ x /| |                            2*|2 - -----| *\x + \/ x /   2*|2 + -----|*|2 - -----||                  /            /      1  \ /      ___\\                                                                                    /            /      1  \ /      ___\\                                                                       /            /      1  \ /      ___\\|
                      |       |            |2 - -----|*\x + \/ x /|                                                        |                  ___         |      ___|                  |      ___| |      ___||                 |                  ___         |      ___|                  |      ___| |      ___||     |            |2 - -----|*\x + \/ x /|                  |            |2 - -----|*\x + \/ x /|     |            |2 - -----|*\x + \/ x /|     |            |      ___|            | |                  ___         |      ___|                  |      ___| |      ___||                  |            |2 - -----|*\x + \/ x /|                                                                                    |            |2 - -----|*\x + \/ x /|                                                                       |            |2 - -----|*\x + \/ x /||
                      |       |            |      ___|            |                                            /      1  \ |   1        x + \/ x          \    \/ x /                  \    \/ x / \    \/ x /|     /      1  \ |   1        x + \/ x          \    \/ x /                  \    \/ x / \    \/ x /|     |            |      ___|            |                  |            |      ___|            |     |            |      ___|            |     |      1     \    \/ x /            | |   1        x + \/ x          \    \/ x /                  \    \/ x / \    \/ x /|                2 |            |      ___|            |                                    2                             3                 |            |      ___|            |                                                                       |            |      ___|            ||
                      |       |      1     \    \/ x /            |                           /      1  \    8*|2 + -----|*|- ---- + ---------------- + -------------------------- + -------------------------|   8*|2 - -----|*|- ---- + ---------------- + -------------------------- + -------------------------|     |      1     \    \/ x /            |  /      1  \     |      1     \    \/ x /            |     |      1     \    \/ x /            |   6*|2 + ----- + -----------------------|*|- ---- + ---------------- + -------------------------- + -------------------------|     /      1  \  |      1     \    \/ x /            |       /      1  \       /      1  \  /      1  \      /      1  \  /      ___\     |      1     \    \/ x /            |  /      1  \      /      1  \ /      ___\     /      1  \ /      1  \ |      1     \    \/ x /            ||
      _______________ |       |2 + ----- + -----------------------|                        12*|2 - -----|      |      ___| |   3/2    3/2 /  ___    \                     2                    ___            |     |      ___| |   3/2    3/2 /  ___    \                     2                    ___            |   6*|2 + ----- + -----------------------| *|2 + -----|   4*|2 + ----- + -----------------------|   4*|2 + ----- + -----------------------|     |      ___            ___           | |   3/2    3/2 /  ___    \                     2                    ___            |   8*|2 + -----| *|2 + ----- + -----------------------|    12*|2 + -----|    24*|2 - -----| *|2 + -----|   24*|2 - -----| *\x + \/ x /   6*|2 + ----- + -----------------------| *|2 - -----|   24*|2 - -----|*\x + \/ x /   8*|2 + -----|*|2 - -----|*|2 + ----- + -----------------------||
     /  /      ___\   |       |      ___            ___           |        /      ___\        |      ___|      \    \/ x / |  x      x   *\\/ x  - x/          /  ___    \                   \/ x  - x        |     \    \/ x / |  x      x   *\\/ x  - x/          /  ___    \                   \/ x  - x        |     |      ___            ___           |  |      ___|     |      ___            ___           |     |      ___            ___           |     \    \/ x           \/ x  - x       / |  x      x   *\\/ x  - x/          /  ___    \                   \/ x  - x        |     |      ___|  |      ___            ___           |       |      ___|       |      ___|  |      ___|      |      ___|                  |      ___            ___           |  |      ___|      |      ___|                 |      ___| |      ___| |      ___            ___           ||
    /  -\x + \/ x /   | 12    \    \/ x           \/ x  - x       /     12*\x + \/ x /        \    \/ x /                  \                                   \\/ x  - x/                                    /                 \                                   \\/ x  - x/                                    /     \    \/ x           \/ x  - x       /  \    \/ x /     \    \/ x           \/ x  - x       /     \    \/ x           \/ x  - x       /                                           \                                   \\/ x  - x/                                    /     \    \/ x /  \    \/ x           \/ x  - x       /       \    \/ x /       \    \/ x /  \    \/ x /      \    \/ x /                  \    \/ x           \/ x  - x       /  \    \/ x /      \    \/ x /                 \    \/ x / \    \/ x / \    \/ x           \/ x  - x       /|
   /   ------------- *|---- + -------------------------------------- - ---------------- - ---------------- - -------------------------------------------------------------------------------------------------- - -------------------------------------------------------------------------------------------------- - ---------------------------------------------------- - --------------------------------------- + --------------------------------------- + ---------------------------------------------------------------------------------------------------------------------------- + ---------------------------------------------------- + ---------------- + --------------------------- + --------------------------- - ---------------------------------------------------- + -------------------------- + ---------------------------------------------------------------|
  /        ___        | 5/2                           2                 5/2 /  ___    \    3/2 /  ___    \                                                     ___                                                                                              ___                                                                                   2                                    3/2 /  ___    \                           3/2 /      ___\                                                                             ___                                                                                            2                        3/2 /  ___    \                      2                             3                        /      ___\ /  ___    \                                      2                            /      ___\ /  ___    \                    |
\/       \/ x  - x    |x                   /      ___\                 x   *\\/ x  - x/   x   *\\/ x  - x/                                               x + \/ x                                                                                             \/ x  - x                                                                    /      ___\                                    x   *\\/ x  - x/                          x   *\x + \/ x /                                                                       x + \/ x                                                                                  /      ___\                        x   *\\/ x  - x/           /  ___    \                   /  ___    \                         \x + \/ x /*\\/ x  - x/                       3/2 /  ___    \                             \x + \/ x /*\\/ x  - x/                    |
                      \                    \x + \/ x /                                                                                                                                                                                                                                                                                     \x + \/ x /                                                                                                                                                                                                                                                               \x + \/ x /                                                   \\/ x  - x/                   \\/ x  - x/                                                                      x   *\\/ x  - x/                                                                        /
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                                                                                                                                                                                                                                                                                                                                                                                                                                          /      ___\                                                                                                                                                                                                                                                                                                                                                                                                                                      
                                                                                                                                                                                                                                                                                                                                                                                                                                       64*\x + \/ x /                                                                                                                                                                                                                                                                                                                                                                                                                                      
$$\frac{\sqrt{- \frac{\sqrt{x} + x}{\sqrt{x} - x}} \left(\frac{24 \left(2 - \frac{1}{\sqrt{x}}\right)^{3} \left(\sqrt{x} + x\right)}{\left(\sqrt{x} - x\right)^{3}} + \frac{24 \left(2 - \frac{1}{\sqrt{x}}\right)^{2} \left(2 + \frac{1}{\sqrt{x}}\right)}{\left(\sqrt{x} - x\right)^{2}} + \frac{8 \left(2 - \frac{1}{\sqrt{x}}\right) \left(2 + \frac{1}{\sqrt{x}}\right) \left(\frac{\left(2 - \frac{1}{\sqrt{x}}\right) \left(\sqrt{x} + x\right)}{\sqrt{x} - x} + 2 + \frac{1}{\sqrt{x}}\right)}{\left(\sqrt{x} - x\right) \left(\sqrt{x} + x\right)} - \frac{8 \left(2 - \frac{1}{\sqrt{x}}\right) \left(\frac{2 \left(2 - \frac{1}{\sqrt{x}}\right)^{2} \left(\sqrt{x} + x\right)}{\left(\sqrt{x} - x\right)^{2}} + \frac{2 \left(2 - \frac{1}{\sqrt{x}}\right) \left(2 + \frac{1}{\sqrt{x}}\right)}{\sqrt{x} - x} - \frac{1}{x^{\frac{3}{2}}} + \frac{\sqrt{x} + x}{x^{\frac{3}{2}} \left(\sqrt{x} - x\right)}\right)}{\sqrt{x} - x} - \frac{6 \left(2 - \frac{1}{\sqrt{x}}\right) \left(\frac{\left(2 - \frac{1}{\sqrt{x}}\right) \left(\sqrt{x} + x\right)}{\sqrt{x} - x} + 2 + \frac{1}{\sqrt{x}}\right)^{2}}{\left(\sqrt{x} - x\right) \left(\sqrt{x} + x\right)} + \frac{8 \left(2 + \frac{1}{\sqrt{x}}\right)^{2} \left(\frac{\left(2 - \frac{1}{\sqrt{x}}\right) \left(\sqrt{x} + x\right)}{\sqrt{x} - x} + 2 + \frac{1}{\sqrt{x}}\right)}{\left(\sqrt{x} + x\right)^{2}} - \frac{8 \left(2 + \frac{1}{\sqrt{x}}\right) \left(\frac{2 \left(2 - \frac{1}{\sqrt{x}}\right)^{2} \left(\sqrt{x} + x\right)}{\left(\sqrt{x} - x\right)^{2}} + \frac{2 \left(2 - \frac{1}{\sqrt{x}}\right) \left(2 + \frac{1}{\sqrt{x}}\right)}{\sqrt{x} - x} - \frac{1}{x^{\frac{3}{2}}} + \frac{\sqrt{x} + x}{x^{\frac{3}{2}} \left(\sqrt{x} - x\right)}\right)}{\sqrt{x} + x} - \frac{6 \left(2 + \frac{1}{\sqrt{x}}\right) \left(\frac{\left(2 - \frac{1}{\sqrt{x}}\right) \left(\sqrt{x} + x\right)}{\sqrt{x} - x} + 2 + \frac{1}{\sqrt{x}}\right)^{2}}{\left(\sqrt{x} + x\right)^{2}} + \frac{6 \left(\frac{\left(2 - \frac{1}{\sqrt{x}}\right) \left(\sqrt{x} + x\right)}{\sqrt{x} - x} + 2 + \frac{1}{\sqrt{x}}\right) \left(\frac{2 \left(2 - \frac{1}{\sqrt{x}}\right)^{2} \left(\sqrt{x} + x\right)}{\left(\sqrt{x} - x\right)^{2}} + \frac{2 \left(2 - \frac{1}{\sqrt{x}}\right) \left(2 + \frac{1}{\sqrt{x}}\right)}{\sqrt{x} - x} - \frac{1}{x^{\frac{3}{2}}} + \frac{\sqrt{x} + x}{x^{\frac{3}{2}} \left(\sqrt{x} - x\right)}\right)}{\sqrt{x} + x} + \frac{\left(\frac{\left(2 - \frac{1}{\sqrt{x}}\right) \left(\sqrt{x} + x\right)}{\sqrt{x} - x} + 2 + \frac{1}{\sqrt{x}}\right)^{3}}{\left(\sqrt{x} + x\right)^{2}} - \frac{12 \left(2 - \frac{1}{\sqrt{x}}\right)}{x^{\frac{3}{2}} \left(\sqrt{x} - x\right)} + \frac{24 \left(2 - \frac{1}{\sqrt{x}}\right) \left(\sqrt{x} + x\right)}{x^{\frac{3}{2}} \left(\sqrt{x} - x\right)^{2}} + \frac{12 \left(2 + \frac{1}{\sqrt{x}}\right)}{x^{\frac{3}{2}} \left(\sqrt{x} - x\right)} + \frac{4 \left(\frac{\left(2 - \frac{1}{\sqrt{x}}\right) \left(\sqrt{x} + x\right)}{\sqrt{x} - x} + 2 + \frac{1}{\sqrt{x}}\right)}{x^{\frac{3}{2}} \left(\sqrt{x} + x\right)} - \frac{4 \left(\frac{\left(2 - \frac{1}{\sqrt{x}}\right) \left(\sqrt{x} + x\right)}{\sqrt{x} - x} + 2 + \frac{1}{\sqrt{x}}\right)}{x^{\frac{3}{2}} \left(\sqrt{x} - x\right)} + \frac{12}{x^{\frac{5}{2}}} - \frac{12 \left(\sqrt{x} + x\right)}{x^{\frac{5}{2}} \left(\sqrt{x} - x\right)}\right)}{64 \left(\sqrt{x} + x\right)}$$
Gráfico
Derivada de ((x+x^1/2)/(x-x^1/2))^1/2