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

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

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

Gráfico:

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Definida a trozos:

Solución

Ha introducido [src] 
x*sin(x)
--------
 sin(x)
xsin⁡(x)sin⁡(x)\frac{x \sin{\left(x \right)}}{\sin{\left(x \right)}} 
(x*sin(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)=xsin⁡(x)f{\left(x \right)} = x \sin{\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)=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. 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:

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

  2. Simplificamos:

    11

Respuesta:

11

Gráfica
02468-8-6-4-2-1010-2020
Trazar el gráfico f(x)
Trazar el gráfico f'(x)
Primera derivada [src] 
x*cos(x) + sin(x)   x*cos(x)
----------------- - --------
      sin(x)         sin(x)
−xcos⁡(x)sin⁡(x)+xcos⁡(x)+sin⁡(x)sin⁡(x)- \frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{x \cos{\left(x \right)} + \sin{\left(x \right)}}{\sin{\left(x \right)}}
Simplificar
Segunda derivada [src] 
  /         2   \                                                      
  |    2*cos (x)|   -2*cos(x) + x*sin(x)   2*(x*cos(x) + sin(x))*cos(x)
x*|1 + ---------| - -------------------- - ----------------------------
  |        2    |          sin(x)                       2              
  \     sin (x) /                                    sin (x)
x(1+2cos⁡2(x)sin⁡2(x))−xsin⁡(x)−2cos⁡(x)sin⁡(x)−2(xcos⁡(x)+sin⁡(x))cos⁡(x)sin⁡2(x)x \left(1 + \frac{2 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) - \frac{x \sin{\left(x \right)} - 2 \cos{\left(x \right)}}{\sin{\left(x \right)}} - \frac{2 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \cos{\left(x \right)}}{\sin^{2}{\left(x \right)}}
Simplificar
Tercera derivada [src] 
                         /         2   \                         /         2   \                                         
                         |    2*cos (x)|                         |    6*cos (x)|          3*(-2*cos(x) + x*sin(x))*cos(x)
-3*sin(x) - x*cos(x) + 3*|1 + ---------|*(x*cos(x) + sin(x)) - x*|5 + ---------|*cos(x) + -------------------------------
                         |        2    |                         |        2    |                       sin(x)            
                         \     sin (x) /                         \     sin (x) /                                         
-------------------------------------------------------------------------------------------------------------------------
                                                          sin(x)
−x(5+6cos⁡2(x)sin⁡2(x))cos⁡(x)−xcos⁡(x)+3(1+2cos⁡2(x)sin⁡2(x))(xcos⁡(x)+sin⁡(x))+3(xsin⁡(x)−2cos⁡(x))cos⁡(x)sin⁡(x)−3sin⁡(x)sin⁡(x)\frac{- x \left(5 + \frac{6 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \cos{\left(x \right)} - x \cos{\left(x \right)} + 3 \left(1 + \frac{2 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) + \frac{3 \left(x \sin{\left(x \right)} - 2 \cos{\left(x \right)}\right) \cos{\left(x \right)}}{\sin{\left(x \right)}} - 3 \sin{\left(x \right)}}{\sin{\left(x \right)}}
Simplificar
Gráfico
Derivada de x*sin(x)/sin(x)
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