Derivada de y=3*sqrt((x+1)x/(sin*2x))

Ha introducido [src]

      ___________
     / (x + 1)*x 
3*  /  --------- 
  \/    sin(2*x)

3 \sqrt{\frac{x \left(x + 1\right)}{\sin{\left(2 x \right)}}}

3*sqrt(((x + 1)*x)/sin(2*x))

Solución detallada

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

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

Respuesta:

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

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

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

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

Simplificar

Segunda derivada [src]

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

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

Simplificar

Tercera derivada [src]

                   /                                                                               /                                                                              2             \     /                                                                              2             \                                                                                                                 /                                                                              2             \                                                                                                                                                             /                                                                              2             \                                                                                                                                                                                 /                                                             2             \                                               \
                   |                                                                               |                  x*cos(2*x)   (1 + x)*cos(2*x)   (1 + 2*x)*cos(2*x)   4*x*cos (2*x)*(1 + x)|     |                  x*cos(2*x)   (1 + x)*cos(2*x)   (1 + 2*x)*cos(2*x)   4*x*cos (2*x)*(1 + x)|                                                                                                                 |                  x*cos(2*x)   (1 + x)*cos(2*x)   (1 + 2*x)*cos(2*x)   4*x*cos (2*x)*(1 + x)|                                            3                                     2                                     2   /          2*x*(1 + x)*cos(2*x)\ |                  x*cos(2*x)   (1 + x)*cos(2*x)   (1 + 2*x)*cos(2*x)   4*x*cos (2*x)*(1 + x)|                                                                                                                                                /          2*x*(1 + x)*cos(2*x)\ |                  2*x*cos(2*x)   2*(1 + x)*cos(2*x)   4*x*cos (2*x)*(1 + x)|                                     2         |
                   |                     2*x*(1 + x)*cos(2*x)             2*x*(1 + x)*cos(2*x)   2*|1 + 2*x*(1 + x) - ---------- - ---------------- - ------------------ + ---------------------|   2*|1 + 2*x*(1 + x) - ---------- - ---------------- - ------------------ + ---------------------|                          2*x*(1 + x)*cos(2*x)                                                                 4*|1 + 2*x*(1 + x) - ---------- - ---------------- - ------------------ + ---------------------|*cos(2*x)   /          2*x*(1 + x)*cos(2*x)\      /          2*x*(1 + x)*cos(2*x)\      /          2*x*(1 + x)*cos(2*x)\    |1 + 2*x - --------------------|*|1 + 2*x*(1 + x) - ---------- - ---------------- - ------------------ + ---------------------|     /          2*x*(1 + x)*cos(2*x)\              /          2*x*(1 + x)*cos(2*x)\                                                             |1 + 2*x - --------------------|*|1 + 2*x*(1 + x) - ------------ - ------------------ + ---------------------|     /          2*x*(1 + x)*cos(2*x)\          |
       ___________ |           1 + 2*x - --------------------   1 + 2*x - --------------------     |                   sin(2*x)        sin(2*x)            sin(2*x)                 2           |     |                   sin(2*x)        sin(2*x)            sin(2*x)                 2           |                1 + 2*x - --------------------          2             2                     2                    |                   sin(2*x)        sin(2*x)            sin(2*x)                 2           |            |1 + 2*x - --------------------|    3*|1 + 2*x - --------------------|    3*|1 + 2*x - --------------------|    \                sin(2*x)      / |                   sin(2*x)        sin(2*x)            sin(2*x)                 2           |   2*|1 + 2*x - --------------------|*cos(2*x)   2*|1 + 2*x - --------------------|*cos(2*x)                                   3                \                sin(2*x)      / |                    sin(2*x)          sin(2*x)                 2           |   3*|1 + 2*x - --------------------| *cos(2*x)|
      / x*(1 + x)  |                           sin(2*x)                         sin(2*x)           \                                                                             sin (2*x)      /     \                                                                             sin (2*x)      /   6*cos(2*x)                   sin(2*x)         8*x*cos (2*x)   8*cos (2*x)*(1 + x)   4*cos (2*x)*(1 + 2*x)     \                                                                             sin (2*x)      /            \                sin(2*x)      /      \                sin(2*x)      /      \                sin(2*x)      /                                     \                                                                             sin (2*x)      /     \                sin(2*x)      /              \                sin(2*x)      /            16*x*(1 + x)*cos(2*x)   24*x*cos (2*x)*(1 + x)                                    \                                                            sin (2*x)      /     \                sin(2*x)      /          |
-3*  /  --------- *|-4 - 8*x - ------------------------------ - ------------------------------ + ------------------------------------------------------------------------------------------------ + ------------------------------------------------------------------------------------------------ + ---------- - ------------------------------ - ------------- - ------------------- - --------------------- - --------------------------------------------------------------------------------------------------------- - --------------------------------- + ----------------------------------- + ----------------------------------- - ------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------- + ------------------------------------------- + --------------------- + ---------------------- - -------------------------------------------------------------------------------------------------------------- - --------------------------------------------|
   \/    sin(2*x)  |                          2                                   2                                                             x                                                                                                1 + x                                                  sin(2*x)              x*(1 + x)                   2                  2                      2                                                               sin(2*x)                                                                2        2                                    2                             2                                                                                 x*(1 + x)                                                                               x*sin(2*x)                                 (1 + x)*sin(2*x)                       sin(2*x)                  3                                                                2*x*(1 + x)                                                                 2*x*(1 + x)*sin(2*x)            |
                   \                         x                             (1 + x)                                                                                                                                                                                                                                                                     sin (2*x)          sin (2*x)              sin (2*x)                                                                                                                               8*x *(1 + x)                          4*x*(1 + x)                           4*x *(1 + x)                                                                                                                                                                                                                                                                          sin (2*x)                                                                                                                                                                       /
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         x*(1 + x)

- \frac{3 \sqrt{\frac{x \left(x + 1\right)}{\sin{\left(2 x \right)}}} \left(\frac{16 x \left(x + 1\right) \cos{\left(2 x \right)}}{\sin{\left(2 x \right)}} + \frac{24 x \left(x + 1\right) \cos^{3}{\left(2 x \right)}}{\sin^{3}{\left(2 x \right)}} - 8 x - \frac{8 x \cos^{2}{\left(2 x \right)}}{\sin^{2}{\left(2 x \right)}} - \frac{8 \left(x + 1\right) \cos^{2}{\left(2 x \right)}}{\sin^{2}{\left(2 x \right)}} - \frac{4 \left(2 x + 1\right) \cos^{2}{\left(2 x \right)}}{\sin^{2}{\left(2 x \right)}} - \frac{4 \left(2 x \left(x + 1\right) + \frac{4 x \left(x + 1\right) \cos^{2}{\left(2 x \right)}}{\sin^{2}{\left(2 x \right)}} - \frac{x \cos{\left(2 x \right)}}{\sin{\left(2 x \right)}} - \frac{\left(x + 1\right) \cos{\left(2 x \right)}}{\sin{\left(2 x \right)}} - \frac{\left(2 x + 1\right) \cos{\left(2 x \right)}}{\sin{\left(2 x \right)}} + 1\right) \cos{\left(2 x \right)}}{\sin{\left(2 x \right)}} - 4 + \frac{6 \cos{\left(2 x \right)}}{\sin{\left(2 x \right)}} + \frac{2 \left(- \frac{2 x \left(x + 1\right) \cos{\left(2 x \right)}}{\sin{\left(2 x \right)}} + 2 x + 1\right) \cos{\left(2 x \right)}}{\left(x + 1\right) \sin{\left(2 x \right)}} + \frac{2 \left(2 x \left(x + 1\right) + \frac{4 x \left(x + 1\right) \cos^{2}{\left(2 x \right)}}{\sin^{2}{\left(2 x \right)}} - \frac{x \cos{\left(2 x \right)}}{\sin{\left(2 x \right)}} - \frac{\left(x + 1\right) \cos{\left(2 x \right)}}{\sin{\left(2 x \right)}} - \frac{\left(2 x + 1\right) \cos{\left(2 x \right)}}{\sin{\left(2 x \right)}} + 1\right)}{x + 1} - \frac{- \frac{2 x \left(x + 1\right) \cos{\left(2 x \right)}}{\sin{\left(2 x \right)}} + 2 x + 1}{\left(x + 1\right)^{2}} + \frac{2 \left(- \frac{2 x \left(x + 1\right) \cos{\left(2 x \right)}}{\sin{\left(2 x \right)}} + 2 x + 1\right) \cos{\left(2 x \right)}}{x \sin{\left(2 x \right)}} + \frac{2 \left(2 x \left(x + 1\right) + \frac{4 x \left(x + 1\right) \cos^{2}{\left(2 x \right)}}{\sin^{2}{\left(2 x \right)}} - \frac{x \cos{\left(2 x \right)}}{\sin{\left(2 x \right)}} - \frac{\left(x + 1\right) \cos{\left(2 x \right)}}{\sin{\left(2 x \right)}} - \frac{\left(2 x + 1\right) \cos{\left(2 x \right)}}{\sin{\left(2 x \right)}} + 1\right)}{x} - \frac{3 \left(- \frac{2 x \left(x + 1\right) \cos{\left(2 x \right)}}{\sin{\left(2 x \right)}} + 2 x + 1\right)^{2} \cos{\left(2 x \right)}}{2 x \left(x + 1\right) \sin{\left(2 x \right)}} - \frac{\left(- \frac{2 x \left(x + 1\right) \cos{\left(2 x \right)}}{\sin{\left(2 x \right)}} + 2 x + 1\right) \left(2 x \left(x + 1\right) + \frac{4 x \left(x + 1\right) \cos^{2}{\left(2 x \right)}}{\sin^{2}{\left(2 x \right)}} - \frac{2 x \cos{\left(2 x \right)}}{\sin{\left(2 x \right)}} - \frac{2 \left(x + 1\right) \cos{\left(2 x \right)}}{\sin{\left(2 x \right)}} + 1\right)}{2 x \left(x + 1\right)} - \frac{\left(- \frac{2 x \left(x + 1\right) \cos{\left(2 x \right)}}{\sin{\left(2 x \right)}} + 2 x + 1\right) \left(2 x \left(x + 1\right) + \frac{4 x \left(x + 1\right) \cos^{2}{\left(2 x \right)}}{\sin^{2}{\left(2 x \right)}} - \frac{x \cos{\left(2 x \right)}}{\sin{\left(2 x \right)}} - \frac{\left(x + 1\right) \cos{\left(2 x \right)}}{\sin{\left(2 x \right)}} - \frac{\left(2 x + 1\right) \cos{\left(2 x \right)}}{\sin{\left(2 x \right)}} + 1\right)}{x \left(x + 1\right)} - \frac{- \frac{2 x \left(x + 1\right) \cos{\left(2 x \right)}}{\sin{\left(2 x \right)}} + 2 x + 1}{x \left(x + 1\right)} + \frac{3 \left(- \frac{2 x \left(x + 1\right) \cos{\left(2 x \right)}}{\sin{\left(2 x \right)}} + 2 x + 1\right)^{2}}{4 x \left(x + 1\right)^{2}} - \frac{- \frac{2 x \left(x + 1\right) \cos{\left(2 x \right)}}{\sin{\left(2 x \right)}} + 2 x + 1}{x^{2}} + \frac{3 \left(- \frac{2 x \left(x + 1\right) \cos{\left(2 x \right)}}{\sin{\left(2 x \right)}} + 2 x + 1\right)^{2}}{4 x^{2} \left(x + 1\right)} - \frac{\left(- \frac{2 x \left(x + 1\right) \cos{\left(2 x \right)}}{\sin{\left(2 x \right)}} + 2 x + 1\right)^{3}}{8 x^{2} \left(x + 1\right)^{2}}\right)}{x \left(x + 1\right)}

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Derivada de y=3sqrt((x+1)x/(sin2x))

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