Derivada de y=ln(sinxcosx/(e^x)(2^x))

Ha introducido [src]

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

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

log(((sin(x)*cos(x))/E^x)*2^x)

Solución detallada

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

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

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

Respuesta:

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

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

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

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

Simplificar

Segunda derivada [src]

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

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

Simplificar

Tercera derivada [src]

    /   2         2                                          \     /       2           2         2             /   2         2                   \             2                                                         2                 \     /   2         2                                          \     /       2           2         2             /   2         2                   \             2                                                         2                 \      2         2                                                2         2         2    /   2         2                   \                           2       2           2               /       2           2                     \               2                  2       2         3                         2                                               /       2           2         2             /   2         2                   \             2                                                         2                 \     /   2         2                                          \            /   2         2                                          \            /   2         2                                          \            /   2         2                                          \             2    /   2         2                                          \     /   2         2                                          \            /       2           2         2             /   2         2                   \             2                                                         2                 \       
  2*\sin (x) - cos (x) + cos(x)*sin(x) - cos(x)*log(2)*sin(x)/   2*\- 2*sin (x) + 2*cos (x) + sin (x)*log(2) + \sin (x) - cos (x) + cos(x)*sin(x)/*log(2) - cos (x)*log(2) + 3*cos(x)*sin(x) + cos(x)*log(2)*sin(x) - log (2)*cos(x)*sin(x)/   2*\sin (x) - cos (x) + cos(x)*sin(x) - cos(x)*log(2)*sin(x)/   2*\- 2*sin (x) + 2*cos (x) + sin (x)*log(2) + \sin (x) - cos (x) + cos(x)*sin(x)/*log(2) - cos (x)*log(2) + 3*cos(x)*sin(x) + cos(x)*log(2)*sin(x) - log (2)*cos(x)*sin(x)/   sin (x) - cos (x) + cos(x)*sin(x) - cos(x)*log(2)*sin(x)   cos (x) - sin (x) + log (2)*\sin (x) - cos (x) + cos(x)*sin(x)/ - 11*cos(x)*sin(x) - 2*cos (x)*log (2) - 2*sin (x)*log(2) - 2*\- 2*cos (x) + 2*sin (x) - 3*cos(x)*sin(x)/*log(2) + 2*cos (x)*log(2) + 2*log (2)*sin (x) - log (2)*cos(x)*sin(x) + 2*log (2)*cos(x)*sin(x) + 3*cos(x)*log(2)*sin(x)   2*\- 2*sin (x) + 2*cos (x) + sin (x)*log(2) + \sin (x) - cos (x) + cos(x)*sin(x)/*log(2) - cos (x)*log(2) + 3*cos(x)*sin(x) + cos(x)*log(2)*sin(x) - log (2)*cos(x)*sin(x)/   2*\sin (x) - cos (x) + cos(x)*sin(x) - cos(x)*log(2)*sin(x)/*sin(x)   2*\sin (x) - cos (x) + cos(x)*sin(x) - cos(x)*log(2)*sin(x)/*cos(x)   2*\sin (x) - cos (x) + cos(x)*sin(x) - cos(x)*log(2)*sin(x)/*log(2)   2*\sin (x) - cos (x) + cos(x)*sin(x) - cos(x)*log(2)*sin(x)/*log(2)   log (2)*\sin (x) - cos (x) + cos(x)*sin(x) - cos(x)*log(2)*sin(x)/   2*\sin (x) - cos (x) + cos(x)*sin(x) - cos(x)*log(2)*sin(x)/*log(2)   2*\- 2*sin (x) + 2*cos (x) + sin (x)*log(2) + \sin (x) - cos (x) + cos(x)*sin(x)/*log(2) - cos (x)*log(2) + 3*cos(x)*sin(x) + cos(x)*log(2)*sin(x) - log (2)*cos(x)*sin(x)/*log(2)
- ------------------------------------------------------------ - --------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------ + --------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - -------------------------------------------------------- - -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - --------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------- - ------------------------------------------------------------------- - ------------------------------------------------------------------- + ------------------------------------------------------------------- - ------------------------------------------------------------------ + ------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                               2                                                                                                                      2                                                                                                                     2                                                                                                                      2                                                                                                             cos(x)*sin(x)                                                                                                                                                                   cos(x)*sin(x)                                                                                                                                                                                                                             cos(x)*sin(x)                                                                                                                   3                                                                     3                                                                     2                                                                     2                                                              cos(x)*sin(x)                                                         cos(x)*sin(x)                                                                                                                cos(x)*sin(x)                                                                                   
                            cos (x)                                                                                                                cos (x)                                                                                                               sin (x)                                                                                                                sin (x)                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 cos (x)                                                               sin (x)                                                               sin (x)                                                               cos (x)

- \frac{2 \left(\sin^{2}{\left(x \right)} - \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)}}{\cos^{3}{\left(x \right)}} - \frac{2 \left(\sin^{2}{\left(x \right)} - \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} + \frac{2 \left(\sin^{2}{\left(x \right)} - \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \log{\left(2 \right)}}{\cos^{2}{\left(x \right)}} - \frac{\sin^{2}{\left(x \right)} - \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}}{\sin{\left(x \right)} \cos{\left(x \right)}} - \frac{\left(\sin^{2}{\left(x \right)} - \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \log{\left(2 \right)}^{2}}{\sin{\left(x \right)} \cos{\left(x \right)}} + \frac{2 \left(\sin^{2}{\left(x \right)} - \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \log{\left(2 \right)}}{\sin{\left(x \right)} \cos{\left(x \right)}} - \frac{2 \left(\sin^{2}{\left(x \right)} - \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \log{\left(2 \right)}}{\sin^{2}{\left(x \right)}} + \frac{2 \left(\sin^{2}{\left(x \right)} - \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}\right)}{\sin^{2}{\left(x \right)}} - \frac{2 \left(\sin^{2}{\left(x \right)} - \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \cos{\left(x \right)}}{\sin^{3}{\left(x \right)}} - \frac{2 \left(\left(\sin^{2}{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \log{\left(2 \right)} - 2 \sin^{2}{\left(x \right)} + \log{\left(2 \right)} \sin^{2}{\left(x \right)} - \log{\left(2 \right)}^{2} \sin{\left(x \right)} \cos{\left(x \right)} + \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + 3 \sin{\left(x \right)} \cos{\left(x \right)} - \log{\left(2 \right)} \cos^{2}{\left(x \right)} + 2 \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} - \frac{2 \left(\left(\sin^{2}{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \log{\left(2 \right)} - 2 \sin^{2}{\left(x \right)} + \log{\left(2 \right)} \sin^{2}{\left(x \right)} - \log{\left(2 \right)}^{2} \sin{\left(x \right)} \cos{\left(x \right)} + \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + 3 \sin{\left(x \right)} \cos{\left(x \right)} - \log{\left(2 \right)} \cos^{2}{\left(x \right)} + 2 \cos^{2}{\left(x \right)}\right)}{\sin{\left(x \right)} \cos{\left(x \right)}} + \frac{2 \left(\left(\sin^{2}{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \log{\left(2 \right)} - 2 \sin^{2}{\left(x \right)} + \log{\left(2 \right)} \sin^{2}{\left(x \right)} - \log{\left(2 \right)}^{2} \sin{\left(x \right)} \cos{\left(x \right)} + \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + 3 \sin{\left(x \right)} \cos{\left(x \right)} - \log{\left(2 \right)} \cos^{2}{\left(x \right)} + 2 \cos^{2}{\left(x \right)}\right) \log{\left(2 \right)}}{\sin{\left(x \right)} \cos{\left(x \right)}} + \frac{2 \left(\left(\sin^{2}{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \log{\left(2 \right)} - 2 \sin^{2}{\left(x \right)} + \log{\left(2 \right)} \sin^{2}{\left(x \right)} - \log{\left(2 \right)}^{2} \sin{\left(x \right)} \cos{\left(x \right)} + \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + 3 \sin{\left(x \right)} \cos{\left(x \right)} - \log{\left(2 \right)} \cos^{2}{\left(x \right)} + 2 \cos^{2}{\left(x \right)}\right)}{\sin^{2}{\left(x \right)}} - \frac{\left(\sin^{2}{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \log{\left(2 \right)}^{2} - 2 \left(2 \sin^{2}{\left(x \right)} - 3 \sin{\left(x \right)} \cos{\left(x \right)} - 2 \cos^{2}{\left(x \right)}\right) \log{\left(2 \right)} - 2 \log{\left(2 \right)} \sin^{2}{\left(x \right)} - \sin^{2}{\left(x \right)} + 2 \log{\left(2 \right)}^{2} \sin^{2}{\left(x \right)} - 11 \sin{\left(x \right)} \cos{\left(x \right)} - \log{\left(2 \right)}^{3} \sin{\left(x \right)} \cos{\left(x \right)} + 2 \log{\left(2 \right)}^{2} \sin{\left(x \right)} \cos{\left(x \right)} + 3 \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} - 2 \log{\left(2 \right)}^{2} \cos^{2}{\left(x \right)} + \cos^{2}{\left(x \right)} + 2 \log{\left(2 \right)} \cos^{2}{\left(x \right)}}{\sin{\left(x \right)} \cos{\left(x \right)}}

Simplificar

Gráfico

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Derivada de y=ln(sinxcosx/(e^x)(2^x))

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