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log(x)*sin(x)/((3*cos(x)))
Derivada de la función/ log(x)/ sin(x)/ cos(x)/ log(x)*sin(x)/((3*cos(x)))

Derivada de log(x)*sin(x)/((3*cos(x)))

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

Ha introducido [src] 
log(x)*sin(x)
-------------
   3*cos(x)
log⁡(x)sin⁡(x)3cos⁡(x)\frac{\log{\left(x \right)} \sin{\left(x \right)}}{3 \cos{\left(x \right)}} 
(log(x)*sin(x))/((3*cos(x)))
Solución detallada

  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)=log⁡(x)sin⁡(x)f{\left(x \right)} = \log{\left(x \right)} \sin{\left(x \right)} y g(x)=3cos⁡(x)g{\left(x \right)} = 3 \cos{\left(x \right)}.

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

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

      ddxf(x)g(x)=f(x)ddxg(x)+g(x)ddxf(x)\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}

      f(x)=log⁡(x)f{\left(x \right)} = \log{\left(x \right)}; calculamos ddxf(x)\frac{d}{d x} f{\left(x \right)}:

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

      g(x)=sin⁡(x)g{\left(x \right)} = \sin{\left(x \right)}; calculamos ddxg(x)\frac{d}{d x} g{\left(x \right)}:

      1. La derivada del seno es igual al coseno:

        ddxsin⁡(x)=cos⁡(x)\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}

      Como resultado de: log⁡(x)cos⁡(x)+sin⁡(x)x\log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}

    Para calcular ddxg(x)\frac{d}{d x} g{\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. La derivada del coseno es igual a menos el seno:

        ddxcos⁡(x)=−sin⁡(x)\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}

      Entonces, como resultado: −3sin⁡(x)- 3 \sin{\left(x \right)}

    Ahora aplicamos la regla de la derivada de una divesión:

    3(log⁡(x)cos⁡(x)+sin⁡(x)x)cos⁡(x)+3log⁡(x)sin⁡2(x)9cos⁡2(x)\frac{3 \left(\log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}\right) \cos{\left(x \right)} + 3 \log{\left(x \right)} \sin^{2}{\left(x \right)}}{9 \cos^{2}{\left(x \right)}}

  2. Simplificamos:

    xlog⁡(x)+sin⁡(2x)23xcos⁡2(x)\frac{x \log{\left(x \right)} + \frac{\sin{\left(2 x \right)}}{2}}{3 x \cos^{2}{\left(x \right)}}

Respuesta:

xlog⁡(x)+sin⁡(2x)23xcos⁡2(x)\frac{x \log{\left(x \right)} + \frac{\sin{\left(2 x \right)}}{2}}{3 x \cos^{2}{\left(x \right)}}

Gráfica
02468-8-6-4-2-1010-500500
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Primera derivada [src] 
                                       2          
   1     /sin(x)                \   sin (x)*log(x)
--------*|------ + cos(x)*log(x)| + --------------
3*cos(x) \  x                   /          2      
                                      3*cos (x)
(log⁡(x)cos⁡(x)+sin⁡(x)x)13cos⁡(x)+log⁡(x)sin⁡2(x)3cos⁡2(x)\left(\log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}\right) \frac{1}{3 \cos{\left(x \right)}} + \frac{\log{\left(x \right)} \sin^{2}{\left(x \right)}}{3 \cos^{2}{\left(x \right)}}
Simplificar
Segunda derivada [src] 
                                                                        /sin(x)                \       
                                      /         2   \                 2*|------ + cos(x)*log(x)|*sin(x)
  sin(x)                   2*cos(x)   |    2*sin (x)|                   \  x                   /       
- ------ - log(x)*sin(x) + -------- + |1 + ---------|*log(x)*sin(x) + ---------------------------------
     2                        x       |        2    |                               cos(x)             
    x                                 \     cos (x) /                                                  
-------------------------------------------------------------------------------------------------------
                                                3*cos(x)
2(log⁡(x)cos⁡(x)+sin⁡(x)x)sin⁡(x)cos⁡(x)+(2sin⁡2(x)cos⁡2(x)+1)log⁡(x)sin⁡(x)−log⁡(x)sin⁡(x)+2cos⁡(x)x−sin⁡(x)x23cos⁡(x)\frac{\frac{2 \left(\log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}\right) \sin{\left(x \right)}}{\cos{\left(x \right)}} + \left(\frac{2 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) \log{\left(x \right)} \sin{\left(x \right)} - \log{\left(x \right)} \sin{\left(x \right)} + \frac{2 \cos{\left(x \right)}}{x} - \frac{\sin{\left(x \right)}}{x^{2}}}{3 \cos{\left(x \right)}}
Simplificar
Tercera derivada [src] 
                                                                                                                                             /         2   \       
                                                                                        /sin(x)                   2*cos(x)\             2    |    6*sin (x)|       
                                                                                        |------ + log(x)*sin(x) - --------|*sin(x)   sin (x)*|5 + ---------|*log(x)
/         2   \                                                                         |   2                        x    |                  |        2    |       
|    2*sin (x)| /sin(x)                \   sin(x)   cos(x)   cos(x)*log(x)   2*sin(x)   \  x                              /                  \     cos (x) /       
|1 + ---------|*|------ + cos(x)*log(x)| - ------ - ------ - ------------- + -------- - ------------------------------------------ + ------------------------------
|        2    | \  x                   /     x         2           3              3                       cos(x)                                3*cos(x)           
\     cos (x) /                                       x                        3*x                                                                                 
-------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                               cos(x)
(log⁡(x)cos⁡(x)+sin⁡(x)x)(2sin⁡2(x)cos⁡2(x)+1)+(6sin⁡2(x)cos⁡2(x)+5)log⁡(x)sin⁡2(x)3cos⁡(x)−(log⁡(x)sin⁡(x)−2cos⁡(x)x+sin⁡(x)x2)sin⁡(x)cos⁡(x)−log⁡(x)cos⁡(x)3−sin⁡(x)x−cos⁡(x)x2+2sin⁡(x)3x3cos⁡(x)\frac{\left(\log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}\right) \left(\frac{2 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) + \frac{\left(\frac{6 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 5\right) \log{\left(x \right)} \sin^{2}{\left(x \right)}}{3 \cos{\left(x \right)}} - \frac{\left(\log{\left(x \right)} \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)}}{x} + \frac{\sin{\left(x \right)}}{x^{2}}\right) \sin{\left(x \right)}}{\cos{\left(x \right)}} - \frac{\log{\left(x \right)} \cos{\left(x \right)}}{3} - \frac{\sin{\left(x \right)}}{x} - \frac{\cos{\left(x \right)}}{x^{2}} + \frac{2 \sin{\left(x \right)}}{3 x^{3}}}{\cos{\left(x \right)}}
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Gráfico
Derivada de log(x)*sin(x)/((3*cos(x)))
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