Derivada de ((x-sen2x)/(x+sen2x))^2

Ha introducido [src]

              2
/x - sin(2*x)\ 
|------------| 
\x + sin(2*x)/

\left(\frac{x - \sin{\left(2 x \right)}}{x + \sin{\left(2 x \right)}}\right)^{2}

((x - sin(2*x))/(x + sin(2*x)))^2

Solución detallada

Sustituimos $u = \frac{x - \sin{\left(2 x \right)}}{x + \sin{\left(2 x \right)}}$ .
Según el principio, aplicamos: $u^{2}$ tenemos $2 u$
Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \frac{x - \sin{\left(2 x \right)}}{x + \sin{\left(2 x \right)}}$ :
1. Se aplica la regla de la derivada parcial:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = x - \sin{\left(2 x \right)}$ y $g{\left(x \right)} = x + \sin{\left(2 x \right)}$ .
  
  Para calcular $\frac{d}{d x} f{\left(x \right)}$ :
  1. diferenciamos $x - \sin{\left(2 x \right)}$ miembro por miembro:
    1. Según el principio, aplicamos: $x$ tenemos $1$
    2. La derivada del producto de una constante por función es igual al producto de esta constante por la derivada de esta función.
      1. Sustituimos $u = 2 x$ .
      2. La derivada del seno es igual al coseno:
        
        $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
      3. Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} 2 x$ :
        
        La derivada del producto de una constante por función es igual al producto de esta constante por la derivada de esta función.
        
        Según el principio, aplicamos: $x$ tenemos $1$
        
        Entonces, como resultado: $2$
        
        Como resultado de la secuencia de reglas:
        
        $2 \cos{\left(2 x \right)}$
      Entonces, como resultado: $- 2 \cos{\left(2 x \right)}$
    Como resultado de: $1 - 2 \cos{\left(2 x \right)}$
  Para calcular $\frac{d}{d x} g{\left(x \right)}$ :
  1. diferenciamos $x + \sin{\left(2 x \right)}$ miembro por miembro:
    1. Según el principio, aplicamos: $x$ tenemos $1$
    2. Sustituimos $u = 2 x$ .
    3. La derivada del seno es igual al coseno:
      
      $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
    4. Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} 2 x$ :
      1. La derivada del producto de una constante por función es igual al producto de esta constante por la derivada de esta función.
        
        Según el principio, aplicamos: $x$ tenemos $1$
        
        Entonces, como resultado: $2$
      Como resultado de la secuencia de reglas:
      
      $2 \cos{\left(2 x \right)}$
    Como resultado de: $2 \cos{\left(2 x \right)} + 1$
  Ahora aplicamos la regla de la derivada de una divesión:
  
  $\frac{\left(1 - 2 \cos{\left(2 x \right)}\right) \left(x + \sin{\left(2 x \right)}\right) - \left(x - \sin{\left(2 x \right)}\right) \left(2 \cos{\left(2 x \right)} + 1\right)}{\left(x + \sin{\left(2 x \right)}\right)^{2}}$
Como resultado de la secuencia de reglas:

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

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

Respuesta:

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

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

              2                                                                         
(x - sin(2*x))                 /2*(1 - 2*cos(2*x))   2*(-1 - 2*cos(2*x))*(x - sin(2*x))\
---------------*(x + sin(2*x))*|------------------ + ----------------------------------|
              2                |   x + sin(2*x)                             2          |
(x + sin(2*x))                 \                              (x + sin(2*x))           /
----------------------------------------------------------------------------------------
                                      x - sin(2*x)

\frac{\frac{\left(x - \sin{\left(2 x \right)}\right)^{2}}{\left(x + \sin{\left(2 x \right)}\right)^{2}} \left(x + \sin{\left(2 x \right)}\right) \left(\frac{2 \left(1 - 2 \cos{\left(2 x \right)}\right)}{x + \sin{\left(2 x \right)}} + \frac{2 \left(x - \sin{\left(2 x \right)}\right) \left(- 2 \cos{\left(2 x \right)} - 1\right)}{\left(x + \sin{\left(2 x \right)}\right)^{2}}\right)}{x - \sin{\left(2 x \right)}}

Simplificar

Segunda derivada [src]

  /                                                                                                                                                                                                                                                                                                     /                  (1 + 2*cos(2*x))*(x - sin(2*x))\\
  |                                                     2                                                                                            /                             2                                                                                \   (1 + 2*cos(2*x))*(x - sin(2*x))*|-1 + 2*cos(2*x) + -------------------------------||
  |  /                  (1 + 2*cos(2*x))*(x - sin(2*x))\                      /                  (1 + 2*cos(2*x))*(x - sin(2*x))\                    |             (1 + 2*cos(2*x)) *(x - sin(2*x))   (1 + 2*cos(2*x))*(-1 + 2*cos(2*x))   2*(x - sin(2*x))*sin(2*x)|                                   \                            x + sin(2*x)         /|
2*|2*|-1 + 2*cos(2*x) + -------------------------------|  - (-1 + 2*cos(2*x))*|-1 + 2*cos(2*x) + -------------------------------| + 2*(x - sin(2*x))*|2*sin(2*x) + -------------------------------- + ---------------------------------- + -------------------------| - -----------------------------------------------------------------------------------|
  |  \                            x + sin(2*x)         /                      \                            x + sin(2*x)         /                    |                                   2                       x + sin(2*x)                     x + sin(2*x)      |                                       x + sin(2*x)                                   |
  \                                                                                                                                                  \                     (x + sin(2*x))                                                                           /                                                                                      /
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                                                                                                                                                                                    2                                                                                                                                                                       
                                                                                                                                                                      (x + sin(2*x))

\frac{2 \left(2 \left(x - \sin{\left(2 x \right)}\right) \left(\frac{2 \left(x - \sin{\left(2 x \right)}\right) \sin{\left(2 x \right)}}{x + \sin{\left(2 x \right)}} + \frac{\left(x - \sin{\left(2 x \right)}\right) \left(2 \cos{\left(2 x \right)} + 1\right)^{2}}{\left(x + \sin{\left(2 x \right)}\right)^{2}} + 2 \sin{\left(2 x \right)} + \frac{\left(2 \cos{\left(2 x \right)} - 1\right) \left(2 \cos{\left(2 x \right)} + 1\right)}{x + \sin{\left(2 x \right)}}\right) - \frac{\left(x - \sin{\left(2 x \right)}\right) \left(2 \cos{\left(2 x \right)} + 1\right) \left(\frac{\left(x - \sin{\left(2 x \right)}\right) \left(2 \cos{\left(2 x \right)} + 1\right)}{x + \sin{\left(2 x \right)}} + 2 \cos{\left(2 x \right)} - 1\right)}{x + \sin{\left(2 x \right)}} - \left(2 \cos{\left(2 x \right)} - 1\right) \left(\frac{\left(x - \sin{\left(2 x \right)}\right) \left(2 \cos{\left(2 x \right)} + 1\right)}{x + \sin{\left(2 x \right)}} + 2 \cos{\left(2 x \right)} - 1\right) + 2 \left(\frac{\left(x - \sin{\left(2 x \right)}\right) \left(2 \cos{\left(2 x \right)} + 1\right)}{x + \sin{\left(2 x \right)}} + 2 \cos{\left(2 x \right)} - 1\right)^{2}\right)}{\left(x + \sin{\left(2 x \right)}\right)^{2}}

Simplificar

Tercera derivada [src]

  /                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    /                                                                   2               2                   2                                                               \                                                                                                                                                                                                                                                                \
  |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  2                                                                        2                    /                  (1 + 2*cos(2*x))*(x - sin(2*x))\ |                 2                               3*(1 + 2*cos(2*x)) *(x - sin(2*x))    4*(x - sin(2*x)) *sin(2*x)   4*(1 + 2*cos(2*x))*(-1 + 2*cos(2*x))*(x - sin(2*x))|                                                                                                                                                                                                                                                                |
  |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               /                  (1 + 2*cos(2*x))*(x - sin(2*x))\                      /                  (1 + 2*cos(2*x))*(x - sin(2*x))\                     |-1 + 2*cos(2*x) + -------------------------------|*|(-1 + 2*cos(2*x))  + 4*(x - sin(2*x))*sin(2*x) + ----------------------------------- + -------------------------- + ---------------------------------------------------|                   2                /                  (1 + 2*cos(2*x))*(x - sin(2*x))\                                      /                  (1 + 2*cos(2*x))*(x - sin(2*x))\                    /                  (1 + 2*cos(2*x))*(x - sin(2*x))\         |
  |                 /                                                            2                                       3                                                                                                                          \                                                         /                             2                                                                                \                                                                    |-1 + 2*cos(2*x) + -------------------------------| *(-1 + 2*cos(2*x))   |-1 + 2*cos(2*x) + -------------------------------| *(1 + 2*cos(2*x))   \                            x + sin(2*x)         / |                                                                         2                    x + sin(2*x)                              x + sin(2*x)                   |   (1 + 2*cos(2*x)) *(x - sin(2*x))*|-1 + 2*cos(2*x) + -------------------------------|   (1 + 2*cos(2*x))*(-1 + 2*cos(2*x))*|-1 + 2*cos(2*x) + -------------------------------|   2*(x - sin(2*x))*|-1 + 2*cos(2*x) + -------------------------------|*sin(2*x)|
  |                 |              4*(x - sin(2*x))*cos(2*x)   3*(1 + 2*cos(2*x)) *(-1 + 2*cos(2*x))   3*(1 + 2*cos(2*x)) *(x - sin(2*x))   6*(1 + 2*cos(2*x))*sin(2*x)   6*(-1 + 2*cos(2*x))*sin(2*x)   12*(1 + 2*cos(2*x))*(x - sin(2*x))*sin(2*x)|     /                  (1 + 2*cos(2*x))*(x - sin(2*x))\ |             (1 + 2*cos(2*x)) *(x - sin(2*x))   (1 + 2*cos(2*x))*(-1 + 2*cos(2*x))   2*(x - sin(2*x))*sin(2*x)|     /                  (1 + 2*cos(2*x))*(x - sin(2*x))\            \                            x + sin(2*x)         /                      \                            x + sin(2*x)         /                                                                         \                                                           (x + sin(2*x))                                                                                              /                                    \                            x + sin(2*x)         /                                      \                            x + sin(2*x)         /                    \                            x + sin(2*x)         /         |
4*|- (x - sin(2*x))*|-4*cos(2*x) - ------------------------- + ------------------------------------- + ---------------------------------- + --------------------------- + ---------------------------- + -------------------------------------------| - 2*|-1 + 2*cos(2*x) + -------------------------------|*|2*sin(2*x) + -------------------------------- + ---------------------------------- + -------------------------| + 2*|-1 + 2*cos(2*x) + -------------------------------|*sin(2*x) + ---------------------------------------------------------------------- + --------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------ + -------------------------------------------------------------------------------------- + -----------------------------------------------------------------------------|
  |                 |                     x + sin(2*x)                                  2                                     3                     x + sin(2*x)                  x + sin(2*x)                                       2              |     \                            x + sin(2*x)         / |                                   2                       x + sin(2*x)                     x + sin(2*x)      |     \                            x + sin(2*x)         /                                         x - sin(2*x)                                                             x + sin(2*x)                                                                                                                                        x - sin(2*x)                                                                                                                                                           2                                                                           x + sin(2*x)                                                                         x + sin(2*x)                                |
  \                 \                                                     (x + sin(2*x))                        (x + sin(2*x))                                                                                         (x + sin(2*x))               /                                                         \                     (x + sin(2*x))                                                                           /                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       (x + sin(2*x))                                                                                                                                                                                                             /
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                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               2                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 (x + sin(2*x))

\frac{4 \left(- \left(x - \sin{\left(2 x \right)}\right) \left(- \frac{4 \left(x - \sin{\left(2 x \right)}\right) \cos{\left(2 x \right)}}{x + \sin{\left(2 x \right)}} + \frac{12 \left(x - \sin{\left(2 x \right)}\right) \left(2 \cos{\left(2 x \right)} + 1\right) \sin{\left(2 x \right)}}{\left(x + \sin{\left(2 x \right)}\right)^{2}} + \frac{3 \left(x - \sin{\left(2 x \right)}\right) \left(2 \cos{\left(2 x \right)} + 1\right)^{3}}{\left(x + \sin{\left(2 x \right)}\right)^{3}} - 4 \cos{\left(2 x \right)} + \frac{6 \left(2 \cos{\left(2 x \right)} - 1\right) \sin{\left(2 x \right)}}{x + \sin{\left(2 x \right)}} + \frac{6 \left(2 \cos{\left(2 x \right)} + 1\right) \sin{\left(2 x \right)}}{x + \sin{\left(2 x \right)}} + \frac{3 \left(2 \cos{\left(2 x \right)} - 1\right) \left(2 \cos{\left(2 x \right)} + 1\right)^{2}}{\left(x + \sin{\left(2 x \right)}\right)^{2}}\right) + \frac{2 \left(x - \sin{\left(2 x \right)}\right) \left(\frac{\left(x - \sin{\left(2 x \right)}\right) \left(2 \cos{\left(2 x \right)} + 1\right)}{x + \sin{\left(2 x \right)}} + 2 \cos{\left(2 x \right)} - 1\right) \sin{\left(2 x \right)}}{x + \sin{\left(2 x \right)}} + \frac{\left(x - \sin{\left(2 x \right)}\right) \left(2 \cos{\left(2 x \right)} + 1\right)^{2} \left(\frac{\left(x - \sin{\left(2 x \right)}\right) \left(2 \cos{\left(2 x \right)} + 1\right)}{x + \sin{\left(2 x \right)}} + 2 \cos{\left(2 x \right)} - 1\right)}{\left(x + \sin{\left(2 x \right)}\right)^{2}} - 2 \left(\frac{\left(x - \sin{\left(2 x \right)}\right) \left(2 \cos{\left(2 x \right)} + 1\right)}{x + \sin{\left(2 x \right)}} + 2 \cos{\left(2 x \right)} - 1\right) \left(\frac{2 \left(x - \sin{\left(2 x \right)}\right) \sin{\left(2 x \right)}}{x + \sin{\left(2 x \right)}} + \frac{\left(x - \sin{\left(2 x \right)}\right) \left(2 \cos{\left(2 x \right)} + 1\right)^{2}}{\left(x + \sin{\left(2 x \right)}\right)^{2}} + 2 \sin{\left(2 x \right)} + \frac{\left(2 \cos{\left(2 x \right)} - 1\right) \left(2 \cos{\left(2 x \right)} + 1\right)}{x + \sin{\left(2 x \right)}}\right) + 2 \left(\frac{\left(x - \sin{\left(2 x \right)}\right) \left(2 \cos{\left(2 x \right)} + 1\right)}{x + \sin{\left(2 x \right)}} + 2 \cos{\left(2 x \right)} - 1\right) \sin{\left(2 x \right)} + \frac{\left(2 \cos{\left(2 x \right)} - 1\right) \left(2 \cos{\left(2 x \right)} + 1\right) \left(\frac{\left(x - \sin{\left(2 x \right)}\right) \left(2 \cos{\left(2 x \right)} + 1\right)}{x + \sin{\left(2 x \right)}} + 2 \cos{\left(2 x \right)} - 1\right)}{x + \sin{\left(2 x \right)}} + \frac{\left(2 \cos{\left(2 x \right)} + 1\right) \left(\frac{\left(x - \sin{\left(2 x \right)}\right) \left(2 \cos{\left(2 x \right)} + 1\right)}{x + \sin{\left(2 x \right)}} + 2 \cos{\left(2 x \right)} - 1\right)^{2}}{x + \sin{\left(2 x \right)}} + \frac{\left(2 \cos{\left(2 x \right)} - 1\right) \left(\frac{\left(x - \sin{\left(2 x \right)}\right) \left(2 \cos{\left(2 x \right)} + 1\right)}{x + \sin{\left(2 x \right)}} + 2 \cos{\left(2 x \right)} - 1\right)^{2}}{x - \sin{\left(2 x \right)}} - \frac{\left(\frac{\left(x - \sin{\left(2 x \right)}\right) \left(2 \cos{\left(2 x \right)} + 1\right)}{x + \sin{\left(2 x \right)}} + 2 \cos{\left(2 x \right)} - 1\right) \left(\frac{4 \left(x - \sin{\left(2 x \right)}\right)^{2} \sin{\left(2 x \right)}}{x + \sin{\left(2 x \right)}} + \frac{3 \left(x - \sin{\left(2 x \right)}\right)^{2} \left(2 \cos{\left(2 x \right)} + 1\right)^{2}}{\left(x + \sin{\left(2 x \right)}\right)^{2}} + 4 \left(x - \sin{\left(2 x \right)}\right) \sin{\left(2 x \right)} + \frac{4 \left(x - \sin{\left(2 x \right)}\right) \left(2 \cos{\left(2 x \right)} - 1\right) \left(2 \cos{\left(2 x \right)} + 1\right)}{x + \sin{\left(2 x \right)}} + \left(2 \cos{\left(2 x \right)} - 1\right)^{2}\right)}{x - \sin{\left(2 x \right)}}\right)}{\left(x + \sin{\left(2 x \right)}\right)^{2}}

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Derivada de ((x-sen2x)/(x+sen2x))^2

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