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Derivada de:
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Expresiones idénticas
x/(log(x^exp(x))+log(cos(x)/sin(x)))
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Expresiones semejantes
x/(log(x^exp(x))-log(cos(x)/sin(x)))
x/(log(x^exp(x))+log(cosx/sinx))
Expresiones con funciones
Logaritmo log
log(x+cos(x))^(2)
log(y^2-1)
log(x)*sin(x)/((3*cos(x)))
logcosx
log(x+4)^2+2*x+7
Logaritmo log
log(x+cos(x))^(2)
log(y^2-1)
log(x)*sin(x)/((3*cos(x)))
logcosx
log(x+4)^2+2*x+7
Coseno cos
cos(x)*x^3
cos√(sin(tan(πx)))
cose^x
cos(3*x)*e^sin(x)
cos(3)+x^2
Seno sin
sin(x)/cos(x)^(2)
sin(x)^(5)
sin(x)^(4)
sin(x)^(x^2)
sin(x)*sin(x)*sin(x)

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

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

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

Se aplica la regla de la derivada parcial:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = x$ y $g{\left(x \right)} = \log{\left(x^{e^{x}} \right)} + \log{\left(\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} \right)}$ .

Para calcular $\frac{d}{d x} f{\left(x \right)}$ :
1. Según el principio, aplicamos: $x$ tenemos $1$
Para calcular $\frac{d}{d x} g{\left(x \right)}$ :
1. diferenciamos $\log{\left(x^{e^{x}} \right)} + \log{\left(\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} \right)}$ miembro por miembro:
  1. Sustituimos $u = x^{e^{x}}$ .
  2. Derivado $\log{\left(u \right)}$ es $\frac{1}{u}$ .
  3. Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} x^{e^{x}}$ :
    1. No logro encontrar los pasos en la búsqueda de esta derivada.
      
      Perola derivada
      
      $\left(x + 1\right) e^{x e^{x}}$
    Como resultado de la secuencia de reglas:
    
    $x^{- e^{x}} \left(x + 1\right) e^{x e^{x}}$
  4. Sustituimos $u = \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}$ .
  5. Derivado $\log{\left(u \right)}$ es $\frac{1}{u}$ .
  6. Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}$ :
    1. Se aplica la regla de la derivada parcial:
      
      $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
      
      $f{\left(x \right)} = \cos{\left(x \right)}$ y $g{\left(x \right)} = \sin{\left(x \right)}$ .
      
      Para calcular $\frac{d}{d x} f{\left(x \right)}$ :
      1. La derivada del coseno es igual a menos el seno:
        
        $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
      Para calcular $\frac{d}{d x} g{\left(x \right)}$ :
      1. La derivada del seno es igual al coseno:
        
        $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
      Ahora aplicamos la regla de la derivada de una divesión:
      
      $\frac{- \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}$
    Como resultado de la secuencia de reglas:
    
    $\frac{- \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}}{\sin{\left(x \right)} \cos{\left(x \right)}}$
  Como resultado de: $\frac{- \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}}{\sin{\left(x \right)} \cos{\left(x \right)}} + x^{- e^{x}} \left(x + 1\right) e^{x e^{x}}$
Ahora aplicamos la regla de la derivada de una divesión:

$\frac{- x \left(\frac{- \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}}{\sin{\left(x \right)} \cos{\left(x \right)}} + x^{- e^{x}} \left(x + 1\right) e^{x e^{x}}\right) + \log{\left(x^{e^{x}} \right)} + \log{\left(\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} \right)}}{\left(\log{\left(x^{e^{x}} \right)} + \log{\left(\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} \right)}\right)^{2}}$
Simplificamos:

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

Respuesta:

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

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

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)//

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

Simplificar

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)//

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

Simplificar

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}}$$

Simplificar

Gráfico

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

Otras calculadoras

¿Cómo usar?

Expresiones idénticas

Expresiones semejantes

Expresiones con funciones

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

Gráfico:

Definida a trozos:

Solución