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cos(sqrt(4*x/log(x)))

Derivada de cos(sqrt(4*x/log(x)))

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

Gráfico:

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Definida a trozos:

Solución

Ha introducido [src]
   /    ________\
   |   /  4*x   |
cos|  /  ------ |
   \\/   log(x) /
$$\cos{\left(\sqrt{\frac{4 x}{\log{\left(x \right)}}} \right)}$$
cos(sqrt((4*x)/log(x)))
Solución detallada
  1. Sustituimos .

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

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

    1. Sustituimos .

    2. Según el principio, aplicamos: tenemos

    3. 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 producto de una constante por función es igual al producto de esta constante por la derivada de esta función.

          1. Según el principio, aplicamos: tenemos

          Entonces, como resultado:

        Para calcular :

        1. Derivado es .

        Ahora aplicamos la regla de la derivada de una divesión:

      Como resultado de la secuencia de reglas:

    Como resultado de la secuencia de reglas:

  4. Simplificamos:


Respuesta:

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