Sr Examen

Otras calculadoras


x/(log(x^exp(x))+log(cos(x)/sin(x)))

Derivada de x/(log(x^exp(x))+log(cos(x)/sin(x)))

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

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src]
           x            
------------------------
   / / x\\              
   | \e /|      /cos(x)\
log\x    / + log|------|
                \sin(x)/
$$\frac{x}{\log{\left(x^{e^{x}} \right)} + \log{\left(\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} \right)}}$$
x/(log(x^exp(x)) + log(cos(x)/sin(x)))
Solución detallada
  1. Se aplica la regla de la derivada parcial:

    y .

    Para calcular :

    1. Según el principio, aplicamos: tenemos

    Para calcular :

    1. diferenciamos miembro por miembro:

      1. Sustituimos .

      2. Derivado es .

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

        1. No logro encontrar los pasos en la búsqueda de esta derivada.

          Perola derivada

        Como resultado de la secuencia de reglas:

      4. Sustituimos .

      5. Derivado es .

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

        1. Se aplica la regla de la derivada parcial:

          y .

          Para calcular :

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

          Para calcular :

          1. La derivada del seno es igual al coseno:

          Ahora aplicamos la regla de la derivada de una divesión:

        Como resultado de la secuencia de reglas:

      Como resultado de:

    Ahora aplicamos la regla de la derivada de una divesión:

  2. Simplificamos:


Respuesta:

Gráfica
Primera derivada [src]
                             /                   /        2   \       \
                             |                   |     cos (x)|       |
                             |                   |-1 - -------|*sin(x)|
                             |   x               |        2   |       |
                             |  e     x          \     sin (x)/       |
                           x*|- -- - e *log(x) - ---------------------|
           1                 \  x                        cos(x)       /
------------------------ + --------------------------------------------
   / / x\\                                                   2         
   | \e /|      /cos(x)\           /   / / x\\              \          
log\x    / + log|------|           |   | \e /|      /cos(x)\|          
                \sin(x)/           |log\x    / + log|------||          
                                   \                \sin(x)//          
$$\frac{x \left(- \frac{\left(-1 - \frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \sin{\left(x \right)}}{\cos{\left(x \right)}} - e^{x} \log{\left(x \right)} - \frac{e^{x}}{x}\right)}{\left(\log{\left(x^{e^{x}} \right)} + \log{\left(\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} \right)}\right)^{2}} + \frac{1}{\log{\left(x^{e^{x}} \right)} + \log{\left(\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} \right)}}$$
Segunda derivada [src]
 /  /                                                                        2                               \                                              \ 
 |  |                                 /                 /       2   \       \                                |                                              | 
 |  |                                 |                 |    cos (x)|       |                                |                                              | 
 |  |                                 |                 |1 + -------|*sin(x)|                   /       2   \|                          /       2   \       | 
 |  |                                 | x               |       2   |       |              2    |    cos (x)||                          |    cos (x)|       | 
 |  |                                 |e     x          \    sin (x)/       |           sin (x)*|1 + -------||                        2*|1 + -------|*sin(x)| 
 |  |       2                   x   2*|-- + e *log(x) - --------------------|       x           |       2   ||      x                   |       2   |       | 
 |  |    cos (x)    x          e      \x                       cos(x)       /    2*e            \    sin (x)/|   2*e       x            \    sin (x)/       | 
-|x*|1 + ------- + e *log(x) - -- - ------------------------------------------ + ---- - ---------------------| + ---- + 2*e *log(x) - ----------------------| 
 |  |       2                   2               / / x\\                           x               2          |    x                           cos(x)        | 
 |  |    sin (x)               x                | \e /|      /cos(x)\                          cos (x)       |                                              | 
 |  |                                        log\x    / + log|------|                                        |                                              | 
 \  \                                                        \sin(x)/                                        /                                              / 
--------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                           2                                                                  
                                                                 /   / / x\\              \                                                                   
                                                                 |   | \e /|      /cos(x)\|                                                                   
                                                                 |log\x    / + log|------||                                                                   
                                                                 \                \sin(x)//                                                                   
$$- \frac{x \left(- \frac{\left(1 + \frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + e^{x} \log{\left(x \right)} + 1 + \frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} - \frac{2 \left(- \frac{\left(1 + \frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \sin{\left(x \right)}}{\cos{\left(x \right)}} + e^{x} \log{\left(x \right)} + \frac{e^{x}}{x}\right)^{2}}{\log{\left(x^{e^{x}} \right)} + \log{\left(\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} \right)}} + \frac{2 e^{x}}{x} - \frac{e^{x}}{x^{2}}\right) - \frac{2 \left(1 + \frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \sin{\left(x \right)}}{\cos{\left(x \right)}} + 2 e^{x} \log{\left(x \right)} + \frac{2 e^{x}}{x}}{\left(\log{\left(x^{e^{x}} \right)} + \log{\left(\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} \right)}\right)^{2}}$$
Tercera derivada [src]
 /      /                                                                                                 3     /                 /       2   \       \ /                                              /       2   \\                                                                                          \                                            2                                                                  \ 
 |      |                                                          /                 /       2   \       \      |                 |    cos (x)|       | |                                         2    |    cos (x)||                                                                                          |     /                 /       2   \       \                                                                   | 
 |      |                                                          |                 |    cos (x)|       |      |                 |1 + -------|*sin(x)| |                                      sin (x)*|1 + -------||                                                                                          |     |                 |    cos (x)|       |                                                                   | 
 |      |                                                          |                 |1 + -------|*sin(x)|      | x               |       2   |       | |       2                   x      x           |       2   ||     /         4           2   \                    /       2   \     /       2   \       |     |                 |1 + -------|*sin(x)|                                                      /       2   \| 
 |      |                                                          | x               |       2   |       |      |e     x          \    sin (x)/       | |    cos (x)    x          e    2*e            \    sin (x)/|     |    3*cos (x)   4*cos (x)|               3    |    cos (x)|     |    cos (x)|       |     | x               |       2   |       |                                                 2    |    cos (x)|| 
 |      |                                                          |e     x          \    sin (x)/       |    6*|-- + e *log(x) - --------------------|*|1 + ------- + e *log(x) - -- + ---- - ---------------------|   2*|1 + --------- + ---------|*sin(x)   2*sin (x)*|1 + -------|   2*|1 + -------|*sin(x)|     |e     x          \    sin (x)/       |                                            3*sin (x)*|1 + -------|| 
 |      |               x      x      x        3                 6*|-- + e *log(x) - --------------------|      \x                       cos(x)       / |       2                   2    x               2          |     |        4           2    |                    |       2   |     |       2   |       |   6*|-- + e *log(x) - --------------------|       x        2                       x             |       2   || 
 |      | x          3*e    2*e    3*e    4*cos (x)   4*cos(x)     \x                       cos(x)       /                                              \    sin (x)               x                  cos (x)       /     \     sin (x)     sin (x) /                    \    sin (x)/     \    sin (x)/       |     \x                       cos(x)       /    3*e    3*cos (x)      x          6*e              \    sin (x)/| 
-|3 + x*|e *log(x) - ---- + ---- + ---- + --------- + -------- + ------------------------------------------ - ------------------------------------------------------------------------------------------------------- - ------------------------------------ - ----------------------- + ----------------------| - ------------------------------------------ - ---- + --------- + 3*e *log(x) + ---- - -----------------------| 
 |      |              2      3     x         3        sin(x)                                     2                                                      / / x\\                                                                       cos(x)                             3                      cos(x)        |               / / x\\                            2        2                      x                2           | 
 |      |             x      x             sin (x)                      /   / / x\\              \                                                       | \e /|      /cos(x)\                                                                                         cos (x)                                 |               | \e /|      /cos(x)\             x      sin (x)                                 cos (x)        | 
 |      |                                                               |   | \e /|      /cos(x)\|                                                    log\x    / + log|------|                                                                                                                                 |            log\x    / + log|------|                                                                           | 
 |      |                                                               |log\x    / + log|------||                                                                    \sin(x)/                                                                                                                                 |                            \sin(x)/                                                                           | 
 \      \                                                               \                \sin(x)//                                                                                                                                                                                                             /                                                                                                               / 
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                                                                                                             2                                                                                                                                                                                                   
                                                                                                                                                                                                   /   / / x\\              \                                                                                                                                                                                                    
                                                                                                                                                                                                   |   | \e /|      /cos(x)\|                                                                                                                                                                                                    
                                                                                                                                                                                                   |log\x    / + log|------||                                                                                                                                                                                                    
                                                                                                                                                                                                   \                \sin(x)//                                                                                                                                                                                                    
$$- \frac{x \left(- \frac{2 \left(1 + \frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \sin^{3}{\left(x \right)}}{\cos^{3}{\left(x \right)}} + \frac{2 \left(1 + \frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \sin{\left(x \right)}}{\cos{\left(x \right)}} - \frac{2 \left(1 + \frac{4 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \frac{3 \cos^{4}{\left(x \right)}}{\sin^{4}{\left(x \right)}}\right) \sin{\left(x \right)}}{\cos{\left(x \right)}} + e^{x} \log{\left(x \right)} + \frac{4 \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{4 \cos^{3}{\left(x \right)}}{\sin^{3}{\left(x \right)}} - \frac{6 \left(- \frac{\left(1 + \frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \sin{\left(x \right)}}{\cos{\left(x \right)}} + e^{x} \log{\left(x \right)} + \frac{e^{x}}{x}\right) \left(- \frac{\left(1 + \frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + e^{x} \log{\left(x \right)} + 1 + \frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \frac{2 e^{x}}{x} - \frac{e^{x}}{x^{2}}\right)}{\log{\left(x^{e^{x}} \right)} + \log{\left(\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} \right)}} + \frac{6 \left(- \frac{\left(1 + \frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \sin{\left(x \right)}}{\cos{\left(x \right)}} + e^{x} \log{\left(x \right)} + \frac{e^{x}}{x}\right)^{3}}{\left(\log{\left(x^{e^{x}} \right)} + \log{\left(\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} \right)}\right)^{2}} + \frac{3 e^{x}}{x} - \frac{3 e^{x}}{x^{2}} + \frac{2 e^{x}}{x^{3}}\right) - \frac{3 \left(1 + \frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 3 e^{x} \log{\left(x \right)} + 3 + \frac{3 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} - \frac{6 \left(- \frac{\left(1 + \frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \sin{\left(x \right)}}{\cos{\left(x \right)}} + e^{x} \log{\left(x \right)} + \frac{e^{x}}{x}\right)^{2}}{\log{\left(x^{e^{x}} \right)} + \log{\left(\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} \right)}} + \frac{6 e^{x}}{x} - \frac{3 e^{x}}{x^{2}}}{\left(\log{\left(x^{e^{x}} \right)} + \log{\left(\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} \right)}\right)^{2}}$$
Gráfico
Derivada de x/(log(x^exp(x))+log(cos(x)/sin(x)))