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x*sinx/lnx
Derivada de la función/ x*sin/ x/lnx/ x*sinx/lnx

Derivada de x*sinx/lnx

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

Ha introducido [src] 
x*sin(x)
--------
 log(x)
xsin⁡(x)log⁡(x)\frac{x \sin{\left(x \right)}}{\log{\left(x \right)}} 
(x*sin(x))/log(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)=xsin⁡(x)f{\left(x \right)} = x \sin{\left(x \right)} 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. 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)=xf{\left(x \right)} = x; calculamos ddxf(x)\frac{d}{d x} f{\left(x \right)}:

      1. Según el principio, aplicamos: xx tenemos 11

      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: xcos⁡(x)+sin⁡(x)x \cos{\left(x \right)} + \sin{\left(x \right)}

    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:

    (xcos⁡(x)+sin⁡(x))log⁡(x)−sin⁡(x)log⁡(x)2\frac{\left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \log{\left(x \right)} - \sin{\left(x \right)}}{\log{\left(x \right)}^{2}}

Respuesta:

(xcos⁡(x)+sin⁡(x))log⁡(x)−sin⁡(x)log⁡(x)2\frac{\left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \log{\left(x \right)} - \sin{\left(x \right)}}{\log{\left(x \right)}^{2}}

Primera derivada [src] 
x*cos(x) + sin(x)    sin(x)
----------------- - -------
      log(x)           2   
                    log (x)
xcos⁡(x)+sin⁡(x)log⁡(x)−sin⁡(x)log⁡(x)2\frac{x \cos{\left(x \right)} + \sin{\left(x \right)}}{\log{\left(x \right)}} - \frac{\sin{\left(x \right)}}{\log{\left(x \right)}^{2}}
Simplificar
Segunda derivada [src] 
                                              /      2   \       
                                              |1 + ------|*sin(x)
                      2*(x*cos(x) + sin(x))   \    log(x)/       
2*cos(x) - x*sin(x) - --------------------- + -------------------
                             x*log(x)               x*log(x)     
-----------------------------------------------------------------
                              log(x)
−xsin⁡(x)+2cos⁡(x)+(1+2log⁡(x))sin⁡(x)xlog⁡(x)−2(xcos⁡(x)+sin⁡(x))xlog⁡(x)log⁡(x)\frac{- x \sin{\left(x \right)} + 2 \cos{\left(x \right)} + \frac{\left(1 + \frac{2}{\log{\left(x \right)}}\right) \sin{\left(x \right)}}{x \log{\left(x \right)}} - \frac{2 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right)}{x \log{\left(x \right)}}}{\log{\left(x \right)}}
Simplificar
Tercera derivada [src] 
                                                    /      3         3   \                                            
                                                  2*|1 + ------ + -------|*sin(x)     /      2   \                    
                                                    |    log(x)      2   |          3*|1 + ------|*(x*cos(x) + sin(x))
                       3*(-2*cos(x) + x*sin(x))     \             log (x)/            \    log(x)/                    
-3*sin(x) - x*cos(x) + ------------------------ - ------------------------------- + ----------------------------------
                               x*log(x)                       2                                  2                    
                                                             x *log(x)                          x *log(x)             
----------------------------------------------------------------------------------------------------------------------
                                                        log(x)
−xcos⁡(x)−3sin⁡(x)+3(xsin⁡(x)−2cos⁡(x))xlog⁡(x)+3(1+2log⁡(x))(xcos⁡(x)+sin⁡(x))x2log⁡(x)−2(1+3log⁡(x)+3log⁡(x)2)sin⁡(x)x2log⁡(x)log⁡(x)\frac{- x \cos{\left(x \right)} - 3 \sin{\left(x \right)} + \frac{3 \left(x \sin{\left(x \right)} - 2 \cos{\left(x \right)}\right)}{x \log{\left(x \right)}} + \frac{3 \left(1 + \frac{2}{\log{\left(x \right)}}\right) \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right)}{x^{2} \log{\left(x \right)}} - \frac{2 \left(1 + \frac{3}{\log{\left(x \right)}} + \frac{3}{\log{\left(x \right)}^{2}}\right) \sin{\left(x \right)}}{x^{2} \log{\left(x \right)}}}{\log{\left(x \right)}}
Simplificar
Gráfico
Derivada de x*sinx/lnx
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