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y=(ln^3sinx)/cos2x

Derivada de y=(ln^3sinx)/cos2x

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

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Solución

Ha introducido [src]
   3          
log (x)*sin(x)
--------------
   cos(2*x)   
log(x)3sin(x)cos(2x)\frac{\log{\left(x \right)}^{3} \sin{\left(x \right)}}{\cos{\left(2 x \right)}}
(log(x)^3*sin(x))/cos(2*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)3sin(x)f{\left(x \right)} = \log{\left(x \right)}^{3} \sin{\left(x \right)} y g(x)=cos(2x)g{\left(x \right)} = \cos{\left(2 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)3f{\left(x \right)} = \log{\left(x \right)}^{3}; calculamos ddxf(x)\frac{d}{d x} f{\left(x \right)}:

      1. Sustituimos u=log(x)u = \log{\left(x \right)}.

      2. Según el principio, aplicamos: u3u^{3} tenemos 3u23 u^{2}

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

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

        Como resultado de la secuencia de reglas:

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

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

    1. Sustituimos u=2xu = 2 x.

    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 ddx2x\frac{d}{d x} 2 x:

      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: 22

      Como resultado de la secuencia de reglas:

      2sin(2x)- 2 \sin{\left(2 x \right)}

    Ahora aplicamos la regla de la derivada de una divesión:

    (log(x)3cos(x)+3log(x)2sin(x)x)cos(2x)+2log(x)3sin(x)sin(2x)cos2(2x)\frac{\left(\log{\left(x \right)}^{3} \cos{\left(x \right)} + \frac{3 \log{\left(x \right)}^{2} \sin{\left(x \right)}}{x}\right) \cos{\left(2 x \right)} + 2 \log{\left(x \right)}^{3} \sin{\left(x \right)} \sin{\left(2 x \right)}}{\cos^{2}{\left(2 x \right)}}

  2. Simplificamos:

    (2xlog(x)cos3(x)+3xlog(x)cos(x)6sin3(x)+3sin(x))log(x)2xcos2(2x)\frac{\left(- 2 x \log{\left(x \right)} \cos^{3}{\left(x \right)} + 3 x \log{\left(x \right)} \cos{\left(x \right)} - 6 \sin^{3}{\left(x \right)} + 3 \sin{\left(x \right)}\right) \log{\left(x \right)}^{2}}{x \cos^{2}{\left(2 x \right)}}


Respuesta:

(2xlog(x)cos3(x)+3xlog(x)cos(x)6sin3(x)+3sin(x))log(x)2xcos2(2x)\frac{\left(- 2 x \log{\left(x \right)} \cos^{3}{\left(x \right)} + 3 x \log{\left(x \right)} \cos{\left(x \right)} - 6 \sin^{3}{\left(x \right)} + 3 \sin{\left(x \right)}\right) \log{\left(x \right)}^{2}}{x \cos^{2}{\left(2 x \right)}}

Gráfica
02468-8-6-4-2-1010-100005000
Primera derivada [src]
                      2                                      
   3             3*log (x)*sin(x)                            
log (x)*cos(x) + ----------------        3                   
                        x           2*log (x)*sin(x)*sin(2*x)
--------------------------------- + -------------------------
             cos(2*x)                          2             
                                            cos (2*x)        
log(x)3cos(x)+3log(x)2sin(x)xcos(2x)+2log(x)3sin(x)sin(2x)cos2(2x)\frac{\log{\left(x \right)}^{3} \cos{\left(x \right)} + \frac{3 \log{\left(x \right)}^{2} \sin{\left(x \right)}}{x}}{\cos{\left(2 x \right)}} + \frac{2 \log{\left(x \right)}^{3} \sin{\left(x \right)} \sin{\left(2 x \right)}}{\cos^{2}{\left(2 x \right)}}
Segunda derivada [src]
/                                                                                                     /                3*sin(x)\                \       
|                                                      /         2     \                            4*|cos(x)*log(x) + --------|*log(x)*sin(2*x)|       
|     2             3*(-2 + log(x))*sin(x)        2    |    2*sin (2*x)|          6*cos(x)*log(x)     \                   x    /                |       
|- log (x)*sin(x) - ---------------------- + 4*log (x)*|1 + -----------|*sin(x) + --------------- + --------------------------------------------|*log(x)
|                              2                       |        2      |                 x                            cos(2*x)                  |       
\                             x                        \     cos (2*x) /                                                                        /       
--------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                        cos(2*x)                                                                        
(4(log(x)cos(x)+3sin(x)x)log(x)sin(2x)cos(2x)+4(2sin2(2x)cos2(2x)+1)log(x)2sin(x)log(x)2sin(x)+6log(x)cos(x)x3(log(x)2)sin(x)x2)log(x)cos(2x)\frac{\left(\frac{4 \left(\log{\left(x \right)} \cos{\left(x \right)} + \frac{3 \sin{\left(x \right)}}{x}\right) \log{\left(x \right)} \sin{\left(2 x \right)}}{\cos{\left(2 x \right)}} + 4 \left(\frac{2 \sin^{2}{\left(2 x \right)}}{\cos^{2}{\left(2 x \right)}} + 1\right) \log{\left(x \right)}^{2} \sin{\left(x \right)} - \log{\left(x \right)}^{2} \sin{\left(x \right)} + \frac{6 \log{\left(x \right)} \cos{\left(x \right)}}{x} - \frac{3 \left(\log{\left(x \right)} - 2\right) \sin{\left(x \right)}}{x^{2}}\right) \log{\left(x \right)}}{\cos{\left(2 x \right)}}
Tercera derivada [src]
                                                                                                                                                                                                                                                              /         2     \                
                                                                                                                                                                      /   2             6*cos(x)*log(x)   3*(-2 + log(x))*sin(x)\                        3    |    6*sin (2*x)|                
                                                                                                                                                                    6*|log (x)*sin(x) - --------------- + ----------------------|*log(x)*sin(2*x)   8*log (x)*|5 + -----------|*sin(x)*sin(2*x)
                        2               /       2              \                     /         2     \                                                                |                        x                     2          |                             |        2      |                
     3             9*log (x)*sin(x)   6*\1 + log (x) - 3*log(x)/*sin(x)         2    |    2*sin (2*x)| /                3*sin(x)\   9*(-2 + log(x))*cos(x)*log(x)     \                                             x           /                             \     cos (2*x) /                
- log (x)*cos(x) - ---------------- + --------------------------------- + 12*log (x)*|1 + -----------|*|cos(x)*log(x) + --------| - ----------------------------- - ----------------------------------------------------------------------------- + -------------------------------------------
                          x                            3                             |        2      | \                   x    /                  2                                                   cos(2*x)                                                       cos(2*x)                 
                                                      x                              \     cos (2*x) /                                            x                                                                                                                                            
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                            cos(2*x)                                                                                                                                           
12(log(x)cos(x)+3sin(x)x)(2sin2(2x)cos2(2x)+1)log(x)2+8(6sin2(2x)cos2(2x)+5)log(x)3sin(x)sin(2x)cos(2x)6(log(x)2sin(x)6log(x)cos(x)x+3(log(x)2)sin(x)x2)log(x)sin(2x)cos(2x)log(x)3cos(x)9log(x)2sin(x)x9(log(x)2)log(x)cos(x)x2+6(log(x)23log(x)+1)sin(x)x3cos(2x)\frac{12 \left(\log{\left(x \right)} \cos{\left(x \right)} + \frac{3 \sin{\left(x \right)}}{x}\right) \left(\frac{2 \sin^{2}{\left(2 x \right)}}{\cos^{2}{\left(2 x \right)}} + 1\right) \log{\left(x \right)}^{2} + \frac{8 \left(\frac{6 \sin^{2}{\left(2 x \right)}}{\cos^{2}{\left(2 x \right)}} + 5\right) \log{\left(x \right)}^{3} \sin{\left(x \right)} \sin{\left(2 x \right)}}{\cos{\left(2 x \right)}} - \frac{6 \left(\log{\left(x \right)}^{2} \sin{\left(x \right)} - \frac{6 \log{\left(x \right)} \cos{\left(x \right)}}{x} + \frac{3 \left(\log{\left(x \right)} - 2\right) \sin{\left(x \right)}}{x^{2}}\right) \log{\left(x \right)} \sin{\left(2 x \right)}}{\cos{\left(2 x \right)}} - \log{\left(x \right)}^{3} \cos{\left(x \right)} - \frac{9 \log{\left(x \right)}^{2} \sin{\left(x \right)}}{x} - \frac{9 \left(\log{\left(x \right)} - 2\right) \log{\left(x \right)} \cos{\left(x \right)}}{x^{2}} + \frac{6 \left(\log{\left(x \right)}^{2} - 3 \log{\left(x \right)} + 1\right) \sin{\left(x \right)}}{x^{3}}}{\cos{\left(2 x \right)}}
Gráfico
Derivada de y=(ln^3sinx)/cos2x