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x*log(x)/log(10)*sin(x)
Derivada de la función/ x*log/ sin(x)/ log(x)/ x*log(x)/log(10)*sin(x)

Derivada de x*log(x)/log(10)*sin(x)

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

Ha introducido [src] 
x*log(x)       
--------*sin(x)
log(10)
xlog⁡(x)log⁡(10)sin⁡(x)\frac{x \log{\left(x \right)}}{\log{\left(10 \right)}} \sin{\left(x \right)} 
((x*log(x))/log(10))*sin(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)=xlog⁡(x)sin⁡(x)f{\left(x \right)} = x \log{\left(x \right)} \sin{\left(x \right)} y g(x)=log⁡(10)g{\left(x \right)} = \log{\left(10 \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)h(x)=f(x)g(x)ddxh(x)+f(x)h(x)ddxg(x)+g(x)h(x)ddxf(x)\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} h{\left(x \right)} = f{\left(x \right)} g{\left(x \right)} \frac{d}{d x} h{\left(x \right)} + f{\left(x \right)} h{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} h{\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)=log⁡(x)g{\left(x \right)} = \log{\left(x \right)}; calculamos ddxg(x)\frac{d}{d x} g{\left(x \right)}:

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

      h(x)=sin⁡(x)h{\left(x \right)} = \sin{\left(x \right)}; calculamos ddxh(x)\frac{d}{d x} h{\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: xlog⁡(x)cos⁡(x)+log⁡(x)sin⁡(x)+sin⁡(x)x \log{\left(x \right)} \cos{\left(x \right)} + \log{\left(x \right)} \sin{\left(x \right)} + \sin{\left(x \right)}

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

    1. La derivada de una constante log⁡(10)\log{\left(10 \right)} es igual a cero.

    Ahora aplicamos la regla de la derivada de una divesión:

    xlog⁡(x)cos⁡(x)+log⁡(x)sin⁡(x)+sin⁡(x)log⁡(10)\frac{x \log{\left(x \right)} \cos{\left(x \right)} + \log{\left(x \right)} \sin{\left(x \right)} + \sin{\left(x \right)}}{\log{\left(10 \right)}}

Respuesta:

xlog⁡(x)cos⁡(x)+log⁡(x)sin⁡(x)+sin⁡(x)log⁡(10)\frac{x \log{\left(x \right)} \cos{\left(x \right)} + \log{\left(x \right)} \sin{\left(x \right)} + \sin{\left(x \right)}}{\log{\left(10 \right)}}

Gráfica
02468-8-6-4-2-1010-2020
Trazar el gráfico f(x)
Trazar el gráfico f'(x)
Primera derivada [src] 
(1 + log(x))*sin(x)   x*cos(x)*log(x)
------------------- + ---------------
      log(10)             log(10)
xlog⁡(x)cos⁡(x)log⁡(10)+(log⁡(x)+1)sin⁡(x)log⁡(10)\frac{x \log{\left(x \right)} \cos{\left(x \right)}}{\log{\left(10 \right)}} + \frac{\left(\log{\left(x \right)} + 1\right) \sin{\left(x \right)}}{\log{\left(10 \right)}}
Simplificar
Segunda derivada [src] 
sin(x)                                          
------ + 2*(1 + log(x))*cos(x) - x*log(x)*sin(x)
  x                                             
------------------------------------------------
                    log(10)
−xlog⁡(x)sin⁡(x)+2(log⁡(x)+1)cos⁡(x)+sin⁡(x)xlog⁡(10)\frac{- x \log{\left(x \right)} \sin{\left(x \right)} + 2 \left(\log{\left(x \right)} + 1\right) \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}}{\log{\left(10 \right)}}
Simplificar
Tercera derivada [src] 
  sin(x)                           3*cos(x)                  
- ------ - 3*(1 + log(x))*sin(x) + -------- - x*cos(x)*log(x)
     2                                x                      
    x                                                        
-------------------------------------------------------------
                           log(10)
−xlog⁡(x)cos⁡(x)−3(log⁡(x)+1)sin⁡(x)+3cos⁡(x)x−sin⁡(x)x2log⁡(10)\frac{- x \log{\left(x \right)} \cos{\left(x \right)} - 3 \left(\log{\left(x \right)} + 1\right) \sin{\left(x \right)} + \frac{3 \cos{\left(x \right)}}{x} - \frac{\sin{\left(x \right)}}{x^{2}}}{\log{\left(10 \right)}}
Simplificar
Gráfico
Derivada de x*log(x)/log(10)*sin(x)
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