Derivada de y=ln(sinxcosx/e^x2^x)

Ha introducido [src]

   /sin(x)*cos(x)\
log|-------------|
   |     /  x\   |
   |     \x2 /   |
   \    E        /

\log{\left(\frac{\sin{\left(x \right)} \cos{\left(x \right)}}{e^{x_{2}^{x}}} \right)}

log((sin(x)*cos(x))/E^(x2^x))

Solución detallada

Sustituimos $u = \frac{\sin{\left(x \right)} \cos{\left(x \right)}}{e^{x_{2}^{x}}}$ .
Derivado $\log{\left(u \right)}$ es $\frac{1}{u}$ .
Luego se aplica una cadena de reglas. Multiplicamos por $\frac{\partial}{\partial x} \frac{\sin{\left(x \right)} \cos{\left(x \right)}}{e^{x_{2}^{x}}}$ :
1. Se aplica la regla de la derivada parcial:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = \sin{\left(x \right)} \cos{\left(x \right)}$ y $g{\left(x \right)} = e^{x_{2}^{x}}$ .
  
  Para calcular $\frac{d}{d x} f{\left(x \right)}$ :
  1. Se aplica la regla de la derivada de una multiplicación:
    
    $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
    
    $f{\left(x \right)} = \cos{\left(x \right)}$ ; calculamos $\frac{d}{d x} f{\left(x \right)}$ :
    1. La derivada del coseno es igual a menos el seno:
      
      $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
    $g{\left(x \right)} = \sin{\left(x \right)}$ ; calculamos $\frac{d}{d x} g{\left(x \right)}$ :
    1. La derivada del seno es igual al coseno:
      
      $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
    Como resultado de: $- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}$
  Para calcular $\frac{d}{d x} g{\left(x \right)}$ :
  1. Sustituimos $u = x_{2}^{x}$ .
  2. Derivado $e^{u}$ es.
  3. Luego se aplica una cadena de reglas. Multiplicamos por $\frac{\partial}{\partial x} x_{2}^{x}$ :
    1. $\frac{\partial}{\partial x} x_{2}^{x} = x_{2}^{x} \log{\left(x_{2} \right)}$
    Como resultado de la secuencia de reglas:
    
    $x_{2}^{x} e^{x_{2}^{x}} \log{\left(x_{2} \right)}$
  Ahora aplicamos la regla de la derivada de una divesión:
  
  $\left(- x_{2}^{x} e^{x_{2}^{x}} \log{\left(x_{2} \right)} \sin{\left(x \right)} \cos{\left(x \right)} + \left(- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) e^{x_{2}^{x}}\right) e^{- 2 x_{2}^{x}}$
Como resultado de la secuencia de reglas:

$\frac{\left(- x_{2}^{x} e^{x_{2}^{x}} \log{\left(x_{2} \right)} \sin{\left(x \right)} \cos{\left(x \right)} + \left(- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) e^{x_{2}^{x}}\right) e^{- x_{2}^{x}}}{\sin{\left(x \right)} \cos{\left(x \right)}}$
Simplificamos:

$- x_{2}^{x} \log{\left(x_{2} \right)} - 2 \tan{\left(x \right)} + \frac{2}{\sin{\left(2 x \right)}}$

Respuesta:

$- x_{2}^{x} \log{\left(x_{2} \right)} - 2 \tan{\left(x \right)} + \frac{2}{\sin{\left(2 x \right)}}$

Primera derivada [src]

/                        x                  x               \  /  x\
|/   2         2   \  -x2      x         -x2                |  \x2 /
\\cos (x) - sin (x)/*e     - x2 *cos(x)*e    *log(x2)*sin(x)/*e     
--------------------------------------------------------------------
                           cos(x)*sin(x)

\frac{\left(- x_{2}^{x} e^{- x_{2}^{x}} \log{\left(x_{2} \right)} \sin{\left(x \right)} \cos{\left(x \right)} + \left(- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) e^{- x_{2}^{x}}\right) e^{x_{2}^{x}}}{\sin{\left(x \right)} \cos{\left(x \right)}}

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Segunda derivada [src]

   2         2        x                            2         2        x                                              x    2                x /   2         2   \             x    2                2*x    2                       x    2                       x /   2         2        x                      \        
sin (x) - cos (x) + x2 *cos(x)*log(x2)*sin(x)   sin (x) - cos (x) + x2 *cos(x)*log(x2)*sin(x)   -4*cos(x)*sin(x) + x2 *sin (x)*log(x2) + x2 *\sin (x) - cos (x)/*log(x2) - x2 *cos (x)*log(x2) + x2   *log (x2)*cos(x)*sin(x) - x2 *log (x2)*cos(x)*sin(x)   x2 *\sin (x) - cos (x) + x2 *cos(x)*log(x2)*sin(x)/*log(x2)
--------------------------------------------- - --------------------------------------------- + ---------------------------------------------------------------------------------------------------------------------------------------------------------- - -----------------------------------------------------------
                      2                                               2                                                                                               cos(x)*sin(x)                                                                                                 cos(x)*sin(x)                       
                   sin (x)                                         cos (x)

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

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Tercera derivada [src]

    /                     x    2                x /   2         2   \             x    2                2*x    2                       x    2                  \     /                     x    2                x /   2         2   \             x    2                2*x    2                       x    2                  \          2           2        2*x    2     /   2         2   \     x    2     /   2         2   \       x    2        2          2*x    2       2           x    2       2           2*x    2        2        x    3                       3*x    3                          x                             2*x    3                       /   2         2        x                      \            /   2         2        x                      \              x /   2         2        x                      \               x /   2         2        x                      \             x    2     /   2         2        x                      \     2*x    2     /   2         2        x                      \       x /                     x    2                x /   2         2   \             x    2                2*x    2                       x    2                  \        
  2*\-4*cos(x)*sin(x) + x2 *sin (x)*log(x2) + x2 *\sin (x) - cos (x)/*log(x2) - x2 *cos (x)*log(x2) + x2   *log (x2)*cos(x)*sin(x) - x2 *log (x2)*cos(x)*sin(x)/   2*\-4*cos(x)*sin(x) + x2 *sin (x)*log(x2) + x2 *\sin (x) - cos (x)/*log(x2) - x2 *cos (x)*log(x2) + x2   *log (x2)*cos(x)*sin(x) - x2 *log (x2)*cos(x)*sin(x)/   - 4*sin (x) + 4*cos (x) + x2   *log (x2)*\sin (x) - cos (x)/ - x2 *log (x2)*\sin (x) - cos (x)/ - 2*x2 *log (x2)*sin (x) - 2*x2   *cos (x)*log (x2) + 2*x2 *cos (x)*log (x2) + 2*x2   *log (x2)*sin (x) + x2 *log (x2)*cos(x)*sin(x) + x2   *log (x2)*cos(x)*sin(x) - 12*x2 *cos(x)*log(x2)*sin(x) - 3*x2   *log (x2)*cos(x)*sin(x)   2*\sin (x) - cos (x) + x2 *cos(x)*log(x2)*sin(x)/*sin(x)   2*\sin (x) - cos (x) + x2 *cos(x)*log(x2)*sin(x)/*cos(x)   2*x2 *\sin (x) - cos (x) + x2 *cos(x)*log(x2)*sin(x)/*log(x2)   2*x2 *\sin (x) - cos (x) + x2 *cos(x)*log(x2)*sin(x)/*log(x2)   x2 *log (x2)*\sin (x) - cos (x) + x2 *cos(x)*log(x2)*sin(x)/   x2   *log (x2)*\sin (x) - cos (x) + x2 *cos(x)*log(x2)*sin(x)/   2*x2 *\-4*cos(x)*sin(x) + x2 *sin (x)*log(x2) + x2 *\sin (x) - cos (x)/*log(x2) - x2 *cos (x)*log(x2) + x2   *log (x2)*cos(x)*sin(x) - x2 *log (x2)*cos(x)*sin(x)/*log(x2)
- -------------------------------------------------------------------------------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - -------------------------------------------------------- - -------------------------------------------------------- - ------------------------------------------------------------- + ------------------------------------------------------------- - ------------------------------------------------------------ - -------------------------------------------------------------- + --------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                2                                                                                                                                                                2                                                                                                                                                                                                                                             cos(x)*sin(x)                                                                                                                                                                                         3                                                          3                                                             2                                                               2                                                        cos(x)*sin(x)                                                   cos(x)*sin(x)                                                                                                          cos(x)*sin(x)                                                                               
                                                                             sin (x)                                                                                                                                                          cos (x)                                                                                                                                                                                                                                                                                                                                                                                                                                             cos (x)                                                    sin (x)                                                       cos (x)                                                         sin (x)

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

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Derivada de y=ln(sinxcosx/e^x2^x)

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