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x*exp(-x)*sin(x)*cos(x)
Derivada de la función/ x*exp/ sin(x)/ cos(x)/ exp(-x)/ x*exp(-x)*sin(x)*cos(x)

Derivada de x*exp(-x)*sin(x)*cos(x)

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

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Solución

Ha introducido [src] 
   -x              
x*e  *sin(x)*cos(x)
xe−xsin⁡(x)cos⁡(x)x e^{- x} \sin{\left(x \right)} \cos{\left(x \right)} 
((x*exp(-x))*sin(x))*cos(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)cos⁡(x)f{\left(x \right)} = x \sin{\left(x \right)} \cos{\left(x \right)} y g(x)=exg{\left(x \right)} = e^{x}.

    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)=cos⁡(x)g{\left(x \right)} = \cos{\left(x \right)}; calculamos ddxg(x)\frac{d}{d x} g{\left(x \right)}:

      1. La derivada del coseno es igual a menos el seno:

        ddxcos⁡(x)=−sin⁡(x)\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}

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

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

    1. Derivado exe^{x} es.

    Ahora aplicamos la regla de la derivada de una divesión:

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

  2. Simplificamos:

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

Respuesta:

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

Gráfica
02468-8-6-4-2-1010-250000250000
Trazar el gráfico f(x)
Trazar el gráfico f'(x)
Primera derivada [src] 
//     -x    -x\                    -x\               2     -x
\\- x*e   + e  /*sin(x) + x*cos(x)*e  /*cos(x) - x*sin (x)*e
−xe−xsin⁡2(x)+(xe−xcos⁡(x)+(−xe−x+e−x)sin⁡(x))cos⁡(x)- x e^{- x} \sin^{2}{\left(x \right)} + \left(x e^{- x} \cos{\left(x \right)} + \left(- x e^{- x} + e^{- x}\right) \sin{\left(x \right)}\right) \cos{\left(x \right)}
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Segunda derivada [src] 
                                                                                                                      -x
-((x*sin(x) - (-2 + x)*sin(x) + 2*(-1 + x)*cos(x))*cos(x) + 2*(x*cos(x) - (-1 + x)*sin(x))*sin(x) + x*cos(x)*sin(x))*e
−(xsin⁡(x)cos⁡(x)+2(xcos⁡(x)−(x−1)sin⁡(x))sin⁡(x)+(xsin⁡(x)−(x−2)sin⁡(x)+2(x−1)cos⁡(x))cos⁡(x))e−x- \left(x \sin{\left(x \right)} \cos{\left(x \right)} + 2 \left(x \cos{\left(x \right)} - \left(x - 1\right) \sin{\left(x \right)}\right) \sin{\left(x \right)} + \left(x \sin{\left(x \right)} - \left(x - 2\right) \sin{\left(x \right)} + 2 \left(x - 1\right) \cos{\left(x \right)}\right) \cos{\left(x \right)}\right) e^{- x}
Simplificar
Tercera derivada [src] 
/     2                                                                                                                                                                                     \  -x
\x*sin (x) - (x*cos(x) + (-3 + x)*sin(x) - 3*(-1 + x)*sin(x) - 3*(-2 + x)*cos(x))*cos(x) - 3*(x*cos(x) - (-1 + x)*sin(x))*cos(x) + 3*(x*sin(x) - (-2 + x)*sin(x) + 2*(-1 + x)*cos(x))*sin(x)/*e
(xsin⁡2(x)−3(xcos⁡(x)−(x−1)sin⁡(x))cos⁡(x)+3(xsin⁡(x)−(x−2)sin⁡(x)+2(x−1)cos⁡(x))sin⁡(x)−(xcos⁡(x)+(x−3)sin⁡(x)−3(x−2)cos⁡(x)−3(x−1)sin⁡(x))cos⁡(x))e−x\left(x \sin^{2}{\left(x \right)} - 3 \left(x \cos{\left(x \right)} - \left(x - 1\right) \sin{\left(x \right)}\right) \cos{\left(x \right)} + 3 \left(x \sin{\left(x \right)} - \left(x - 2\right) \sin{\left(x \right)} + 2 \left(x - 1\right) \cos{\left(x \right)}\right) \sin{\left(x \right)} - \left(x \cos{\left(x \right)} + \left(x - 3\right) \sin{\left(x \right)} - 3 \left(x - 2\right) \cos{\left(x \right)} - 3 \left(x - 1\right) \sin{\left(x \right)}\right) \cos{\left(x \right)}\right) e^{- x}
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Gráfico
Derivada de x*exp(-x)*sin(x)*cos(x)
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