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

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

Función f() - derivada -er orden en el punto
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Gráfico:

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Solución

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

  1. Sustituimos u=4xlog⁡(x)u = \sqrt{\frac{4 x}{\log{\left(x \right)}}}.

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

    dducos⁡(u)=−sin⁡(u)\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}

  3. Luego se aplica una cadena de reglas. Multiplicamos por ddx4xlog⁡(x)\frac{d}{d x} \sqrt{\frac{4 x}{\log{\left(x \right)}}}:

    1. Sustituimos u=4xlog⁡(x)u = \frac{4 x}{\log{\left(x \right)}}.

    2. Según el principio, aplicamos: u\sqrt{u} tenemos 12u\frac{1}{2 \sqrt{u}}

    3. Luego se aplica una cadena de reglas. Multiplicamos por ddx4xlog⁡(x)\frac{d}{d x} \frac{4 x}{\log{\left(x \right)}}:

      1. Se aplica la regla de la derivada parcial:

        ddxf(x)g(x)=−f(x)ddxg(x)+g(x)ddxf(x)g2(x)\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(x)=4xf{\left(x \right)} = 4 x y g(x)=log⁡(x)g{\left(x \right)} = \log{\left(x \right)}.

        Para calcular ddxf(x)\frac{d}{d x} f{\left(x \right)}:

        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: xx tenemos 11

          Entonces, como resultado: 44

        Para calcular ddxg(x)\frac{d}{d x} g{\left(x \right)}:

        1. Derivado log⁡(x)\log{\left(x \right)} es 1x\frac{1}{x}.

        Ahora aplicamos la regla de la derivada de una divesión:

        4log⁡(x)−4log⁡(x)2\frac{4 \log{\left(x \right)} - 4}{\log{\left(x \right)}^{2}}

      Como resultado de la secuencia de reglas:

      4log⁡(x)−44xlog⁡(x)log⁡(x)2\frac{4 \log{\left(x \right)} - 4}{4 \sqrt{\frac{x}{\log{\left(x \right)}}} \log{\left(x \right)}^{2}}

    Como resultado de la secuencia de reglas:

    −(4log⁡(x)−4)sin⁡(4xlog⁡(x))4xlog⁡(x)log⁡(x)2- \frac{\left(4 \log{\left(x \right)} - 4\right) \sin{\left(\sqrt{\frac{4 x}{\log{\left(x \right)}}} \right)}}{4 \sqrt{\frac{x}{\log{\left(x \right)}}} \log{\left(x \right)}^{2}}

  4. Simplificamos:

    −(log⁡(x)−1)sin⁡(2xlog⁡(x))xlog⁡(x)log⁡(x)2- \frac{\left(\log{\left(x \right)} - 1\right) \sin{\left(2 \sqrt{\frac{x}{\log{\left(x \right)}}} \right)}}{\sqrt{\frac{x}{\log{\left(x \right)}}} \log{\left(x \right)}^{2}}

Respuesta:

−(log⁡(x)−1)sin⁡(2xlog⁡(x))xlog⁡(x)log⁡(x)2- \frac{\left(\log{\left(x \right)} - 1\right) \sin{\left(2 \sqrt{\frac{x}{\log{\left(x \right)}}} \right)}}{\sqrt{\frac{x}{\log{\left(x \right)}}} \log{\left(x \right)}^{2}}

Gráfica
02468-8-6-4-2-1010-1000010000
Trazar el gráfico f(x)
Trazar el gráfico f'(x)
Primera derivada [src] 
       ________                                /    ________\ 
      /   x     /     2        2   \           |   /  4*x   | 
-2*  /  ------ *|- ------- + ------|*log(x)*sin|  /  ------ | 
   \/   log(x)  |     2      log(x)|           \\/   log(x) / 
                \  log (x)         /                          
--------------------------------------------------------------
                             4*x
−2xlog⁡(x)(2log⁡(x)−2log⁡(x)2)log⁡(x)sin⁡(4xlog⁡(x))4x- \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}
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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
−(1−1log⁡(x))2cos⁡(2xlog⁡(x))log⁡(x)+xlog⁡(x)(1−2log⁡(x))sin⁡(2xlog⁡(x))xlog⁡(x)−xlog⁡(x)(1−1log⁡(x))2sin⁡(2xlog⁡(x))2x+xlog⁡(x)(1−1log⁡(x))sin⁡(2xlog⁡(x))x−xlog⁡(x)(1−1log⁡(x))sin⁡(2xlog⁡(x))xlog⁡(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}
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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
xlog⁡(x)(1−1log⁡(x))3sin⁡(2xlog⁡(x))log⁡(x)+3(1−2log⁡(x))(1−1log⁡(x))cos⁡(2xlog⁡(x))log⁡(x)2−3(1−1log⁡(x))3cos⁡(2xlog⁡(x))2log⁡(x)+3(1−1log⁡(x))2cos⁡(2xlog⁡(x))log⁡(x)−3(1−1log⁡(x))2cos⁡(2xlog⁡(x))log⁡(x)2−xlog⁡(x)(1−6log⁡(x)2)sin⁡(2xlog⁡(x))xlog⁡(x)+3xlog⁡(x)(1−2log⁡(x))(1−1log⁡(x))sin⁡(2xlog⁡(x))2xlog⁡(x)−2xlog⁡(x)(1−2log⁡(x))sin⁡(2xlog⁡(x))xlog⁡(x)+2xlog⁡(x)(1−2log⁡(x))sin⁡(2xlog⁡(x))xlog⁡(x)2−xlog⁡(x)(1−1log⁡(x))3sin⁡(2xlog⁡(x))4x+3xlog⁡(x)(1−1log⁡(x))2sin⁡(2xlog⁡(x))2x−3xlog⁡(x)(1−1log⁡(x))2sin⁡(2xlog⁡(x))2xlog⁡(x)−2xlog⁡(x)(1−1log⁡(x))sin⁡(2xlog⁡(x))x+3xlog⁡(x)(1−1log⁡(x))sin⁡(2xlog⁡(x))xlog⁡(x)x2\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}}
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Gráfico
Derivada de cos(sqrt(4*x/log(x)))
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