Derivada de Кореньx/tg(x)

Solución

Ha introducido [src]

    ________
   /   x    
  /  ------ 
\/   tan(x)

$$\sqrt{\frac{x}{\tan{\left(x \right)}}}$$

sqrt(x/tan(x))

Solución detallada

Sustituimos $u = \frac{x}{\tan{\left(x \right)}}$ .
Según el principio, aplicamos: $\sqrt{u}$ tenemos $\frac{1}{2 \sqrt{u}}$
Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \frac{x}{\tan{\left(x \right)}}$ :
1. Se aplica la regla de la derivada parcial:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = x$ y $g{\left(x \right)} = \tan{\left(x \right)}$ .
  
  Para calcular $\frac{d}{d x} f{\left(x \right)}$ :
  1. Según el principio, aplicamos: $x$ tenemos $1$
  Para calcular $\frac{d}{d x} g{\left(x \right)}$ :
  1. Reescribimos las funciones para diferenciar:
    
    $\tan{\left(x \right)} = \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}$
  2. Se aplica la regla de la derivada parcial:
    
    $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
    
    $f{\left(x \right)} = \sin{\left(x \right)}$ y $g{\left(x \right)} = \cos{\left(x \right)}$ .
    
    Para calcular $\frac{d}{d x} f{\left(x \right)}$ :
    1. La derivada del seno es igual al coseno:
      
      $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
    Para calcular $\frac{d}{d x} g{\left(x \right)}$ :
    1. La derivada del coseno es igual a menos el seno:
      
      $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
    Ahora aplicamos la regla de la derivada de una divesión:
    
    $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
  Ahora aplicamos la regla de la derivada de una divesión:
  
  $\frac{- \frac{x \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} + \tan{\left(x \right)}}{\tan^{2}{\left(x \right)}}$
Como resultado de la secuencia de reglas:

$\frac{- \frac{x \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} + \tan{\left(x \right)}}{2 \sqrt{\frac{x}{\tan{\left(x \right)}}} \tan^{2}{\left(x \right)}}$
Simplificamos:

$\frac{- x + \frac{\sin{\left(2 x \right)}}{2}}{\sqrt{\frac{x}{\tan{\left(x \right)}}} \left(1 - \cos{\left(2 x \right)}\right)}$

Respuesta:

$\frac{- x + \frac{\sin{\left(2 x \right)}}{2}}{\sqrt{\frac{x}{\tan{\left(x \right)}}} \left(1 - \cos{\left(2 x \right)}\right)}$

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

    ________ /             /        2   \\       
   /   x     |   1       x*\-1 - tan (x)/|       
  /  ------ *|-------- + ----------------|*tan(x)
\/   tan(x)  |2*tan(x)           2       |       
             \              2*tan (x)    /       
-------------------------------------------------
                        x

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

Simplificar

Segunda derivada [src]

             /                                                                                            2                                       \
             |       /       2   \                                                  /       /       2   \\                  /       /       2   \\|
             |     x*\1 + tan (x)/                                                  |     x*\1 + tan (x)/|    /       2   \ |     x*\1 + tan (x)/||
    ________ |-1 + ---------------                 /               /       2   \\   |-1 + ---------------|    \1 + tan (x)/*|-1 + ---------------||
   /   x     |          tan(x)       /       2   \ |      1      x*\1 + tan (x)/|   \          tan(x)    /                  \          tan(x)    /|
  /  ------ *|-------------------- - \1 + tan (x)/*|x + ------ - ---------------| + ----------------------- - ------------------------------------|
\/   tan(x)  |        2*x                          |    tan(x)          2       |             4*x                           2*tan(x)              |
             \                                     \                 tan (x)    /                                                                 /
---------------------------------------------------------------------------------------------------------------------------------------------------
                                                                         x

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

Simplificar

Tercera derivada [src]

             /                                                                                        2                         3                                                                                                                   2 /               /       2   \\                   /               /       2   \\                                                          /       /       2   \\ /               /       2   \\                           2              \
             |         /       2   \                                            /       /       2   \\    /       /       2   \\                                                                                                       /       2   \  |      1      x*\1 + tan (x)/|     /       2   \ |      1      x*\1 + tan (x)/|                 /       /       2   \\     /       2   \ |     x*\1 + tan (x)/| |      1      x*\1 + tan (x)/|     /       /       2   \\               |
             |       x*\1 + tan (x)/                                            |     x*\1 + tan (x)/|    |     x*\1 + tan (x)/|                  /                                                                      2\          2*\1 + tan (x)/ *|x + ------ - ---------------|   2*\1 + tan (x)/*|x + ------ - ---------------|   /       2   \ |     x*\1 + tan (x)/|   3*\1 + tan (x)/*|-1 + ---------------|*|x + ------ - ---------------|     |     x*\1 + tan (x)/|  /       2   \|
    ________ |  -1 + ---------------                 /       /       2   \\   3*|-1 + ---------------|    |-1 + ---------------|                  |                 /       2   \       /       2   \       /       2   \ |                           |    tan(x)          2       |                   |    tan(x)          2       |   \1 + tan (x)/*|-1 + ---------------|                   \          tan(x)    / |    tan(x)          2       |   3*|-1 + ---------------| *\1 + tan (x)/|
   /   x     |            tan(x)       /       2   \ |     x*\1 + tan (x)/|     \          tan(x)    /    \          tan(x)    /    /       2   \ |        3      3*\1 + tan (x)/   5*x*\1 + tan (x)/   3*x*\1 + tan (x)/ |                           \                 tan (x)    /                   \                 tan (x)    /                 \          tan(x)    /                                          \                 tan (x)    /     \          tan(x)    /               |
  /  ------ *|- -------------------- - \1 + tan (x)/*|-1 + ---------------| - ------------------------- - ----------------------- - \1 + tan (x)/*|2*x + ------ - --------------- - ----------------- + ------------------|*tan(x) - ----------------------------------------------- + ---------------------------------------------- + ------------------------------------ + --------------------------------------------------------------------- + ---------------------------------------|
\/   tan(x)  |            2                          \          tan(x)    /                 2                          2                          |      tan(x)          3                  2                   4         |                               tan(x)                                             x                                        x*tan(x)                                                  2*x                                                   4*x*tan(x)              |
             \           x                                                               4*x                        8*x                           \                   tan (x)            tan (x)             tan (x)      /                                                                                                                                                                                                                                                                   /
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                                                                                                                               x

$$\frac{\sqrt{\frac{x}{\tan{\left(x \right)}}} \left(- \left(\frac{x \left(\tan^{2}{\left(x \right)} + 1\right)}{\tan{\left(x \right)}} - 1\right) \left(\tan^{2}{\left(x \right)} + 1\right) - \frac{2 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \left(- \frac{x \left(\tan^{2}{\left(x \right)} + 1\right)}{\tan^{2}{\left(x \right)}} + x + \frac{1}{\tan{\left(x \right)}}\right)}{\tan{\left(x \right)}} - \left(\tan^{2}{\left(x \right)} + 1\right) \left(\frac{3 x \left(\tan^{2}{\left(x \right)} + 1\right)^{2}}{\tan^{4}{\left(x \right)}} - \frac{5 x \left(\tan^{2}{\left(x \right)} + 1\right)}{\tan^{2}{\left(x \right)}} + 2 x - \frac{3 \left(\tan^{2}{\left(x \right)} + 1\right)}{\tan^{3}{\left(x \right)}} + \frac{3}{\tan{\left(x \right)}}\right) \tan{\left(x \right)} + \frac{3 \left(\frac{x \left(\tan^{2}{\left(x \right)} + 1\right)}{\tan{\left(x \right)}} - 1\right)^{2} \left(\tan^{2}{\left(x \right)} + 1\right)}{4 x \tan{\left(x \right)}} + \frac{3 \left(\frac{x \left(\tan^{2}{\left(x \right)} + 1\right)}{\tan{\left(x \right)}} - 1\right) \left(\tan^{2}{\left(x \right)} + 1\right) \left(- \frac{x \left(\tan^{2}{\left(x \right)} + 1\right)}{\tan^{2}{\left(x \right)}} + x + \frac{1}{\tan{\left(x \right)}}\right)}{2 x} + \frac{\left(\frac{x \left(\tan^{2}{\left(x \right)} + 1\right)}{\tan{\left(x \right)}} - 1\right) \left(\tan^{2}{\left(x \right)} + 1\right)}{x \tan{\left(x \right)}} + \frac{2 \left(\tan^{2}{\left(x \right)} + 1\right) \left(- \frac{x \left(\tan^{2}{\left(x \right)} + 1\right)}{\tan^{2}{\left(x \right)}} + x + \frac{1}{\tan{\left(x \right)}}\right)}{x} - \frac{\left(\frac{x \left(\tan^{2}{\left(x \right)} + 1\right)}{\tan{\left(x \right)}} - 1\right)^{3}}{8 x^{2}} - \frac{3 \left(\frac{x \left(\tan^{2}{\left(x \right)} + 1\right)}{\tan{\left(x \right)}} - 1\right)^{2}}{4 x^{2}} - \frac{\frac{x \left(\tan^{2}{\left(x \right)} + 1\right)}{\tan{\left(x \right)}} - 1}{x^{2}}\right)}{x}$$

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