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

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

Función f() - derivada -er orden en el punto
v

Gráfico:

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Definida a trozos:

Solución

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
  1. Sustituimos .

  2. Derivado es .

  3. Luego se aplica una cadena de reglas. Multiplicamos por :

    1. Se aplica la regla de la derivada parcial:

      y .

      Para calcular :

      1. Se aplica la regla de la derivada de una multiplicación:

        ; calculamos :

        ; calculamos :

        1. La derivada del coseno es igual a menos el seno:

        ; calculamos :

        1. La derivada del seno es igual al coseno:

        Como resultado de:

      Para calcular :

      1. Derivado es.

      Ahora aplicamos la regla de la derivada de una divesión:

    Como resultado de la secuencia de reglas:

  4. Simplificamos:


Respuesta:

Gráfica
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)                                     
$$\frac{2^{- x} \left(2^{x} \left(\left(- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) e^{- x} - e^{- x} \sin{\left(x \right)} \cos{\left(x \right)}\right) + 2^{x} e^{- x} \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)}\right) e^{x}}{\sin{\left(x \right)} \cos{\left(x \right)}}$$
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)                                                                                                                                                                                                                                                                                                                                  
$$- \frac{\sin^{2}{\left(x \right)} - \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} - \frac{\sin^{2}{\left(x \right)} - \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}}{\sin{\left(x \right)} \cos{\left(x \right)}} + \frac{\left(\sin^{2}{\left(x \right)} - \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \log{\left(2 \right)}}{\sin{\left(x \right)} \cos{\left(x \right)}} + \frac{\sin^{2}{\left(x \right)} - \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} - \frac{\left(\sin^{2}{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \log{\left(2 \right)} - 2 \sin^{2}{\left(x \right)} + \log{\left(2 \right)} \sin^{2}{\left(x \right)} - \log{\left(2 \right)}^{2} \sin{\left(x \right)} \cos{\left(x \right)} + \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + 3 \sin{\left(x \right)} \cos{\left(x \right)} - \log{\left(2 \right)} \cos^{2}{\left(x \right)} + 2 \cos^{2}{\left(x \right)}}{\sin{\left(x \right)} \cos{\left(x \right)}}$$
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)}}$$
Gráfico
Derivada de y=ln(sinxcosx/(e^x)(2^x))