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xlnx/sinx
Derivada de la función/ x/sinx/ xlnx/sinx

Derivada de xlnx/sinx

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

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Solución

Ha introducido [src] 
x*log(x)
--------
 sin(x)
xlog⁡(x)sin⁡(x)\frac{x \log{\left(x \right)}}{\sin{\left(x \right)}} 
(x*log(x))/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)f{\left(x \right)} = x \log{\left(x \right)} y g(x)=sin⁡(x)g{\left(x \right)} = \sin{\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)=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}.

      Como resultado de: log⁡(x)+1\log{\left(x \right)} + 1

    Para calcular 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)}

    Ahora aplicamos la regla de la derivada de una divesión:

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

  2. Simplificamos:

    −xlog⁡(x)tan⁡(x)+log⁡(x)+1sin⁡(x)\frac{- \frac{x \log{\left(x \right)}}{\tan{\left(x \right)}} + \log{\left(x \right)} + 1}{\sin{\left(x \right)}}

Respuesta:

−xlog⁡(x)tan⁡(x)+log⁡(x)+1sin⁡(x)\frac{- \frac{x \log{\left(x \right)}}{\tan{\left(x \right)}} + \log{\left(x \right)} + 1}{\sin{\left(x \right)}}

Gráfica
02468-8-6-4-2-1010-5000050000
Trazar el gráfico f(x)
Trazar el gráfico f'(x)
Primera derivada [src] 
1 + log(x)   x*cos(x)*log(x)
---------- - ---------------
  sin(x)            2       
                 sin (x)
−xlog⁡(x)cos⁡(x)sin⁡2(x)+log⁡(x)+1sin⁡(x)- \frac{x \log{\left(x \right)} \cos{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \frac{\log{\left(x \right)} + 1}{\sin{\left(x \right)}}
Simplificar
Segunda derivada [src] 
      /         2   \                               
1     |    2*cos (x)|          2*(1 + log(x))*cos(x)
- + x*|1 + ---------|*log(x) - ---------------------
x     |        2    |                  sin(x)       
      \     sin (x) /                               
----------------------------------------------------
                       sin(x)
x(1+2cos⁡2(x)sin⁡2(x))log⁡(x)−2(log⁡(x)+1)cos⁡(x)sin⁡(x)+1xsin⁡(x)\frac{x \left(1 + \frac{2 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \log{\left(x \right)} - \frac{2 \left(\log{\left(x \right)} + 1\right) \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{1}{x}}{\sin{\left(x \right)}}
Simplificar
Tercera derivada [src] 
                                                     /         2   \              
                                                     |    6*cos (x)|              
                                                   x*|5 + ---------|*cos(x)*log(x)
         /         2   \                             |        2    |              
  1      |    2*cos (x)|                3*cos(x)     \     sin (x) /              
- -- + 3*|1 + ---------|*(1 + log(x)) - -------- - -------------------------------
   2     |        2    |                x*sin(x)                sin(x)            
  x      \     sin (x) /                                                          
----------------------------------------------------------------------------------
                                      sin(x)
−x(5+6cos⁡2(x)sin⁡2(x))log⁡(x)cos⁡(x)sin⁡(x)+3(1+2cos⁡2(x)sin⁡2(x))(log⁡(x)+1)−3cos⁡(x)xsin⁡(x)−1x2sin⁡(x)\frac{- \frac{x \left(5 + \frac{6 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \log{\left(x \right)} \cos{\left(x \right)}}{\sin{\left(x \right)}} + 3 \left(1 + \frac{2 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \left(\log{\left(x \right)} + 1\right) - \frac{3 \cos{\left(x \right)}}{x \sin{\left(x \right)}} - \frac{1}{x^{2}}}{\sin{\left(x \right)}}
Simplificar
Gráfico
Derivada de xlnx/sinx
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