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x^-xsinx
Derivada de la función/ xsinx/ x^-xsinx

Derivada de x^-xsinx

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

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Definida a trozos:

Solución

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

    Para calcular ddxf(x)\frac{d}{d x} f{\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)}

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

    1. No logro encontrar los pasos en la búsqueda de esta derivada.

      Perola derivada

      xx(log⁡(x)+1)x^{x} \left(\log{\left(x \right)} + 1\right)

    Ahora aplicamos la regla de la derivada de una divesión:

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

  2. Simplificamos:

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

Respuesta:

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

Gráfica
02468-8-6-4-2-1010-500000000010000000000
Trazar el gráfico f(x)
Trazar el gráfico f'(x)
Primera derivada [src] 
 -x           -x                     
x  *cos(x) + x  *(-1 - log(x))*sin(x)
x−x(−log⁡(x)−1)sin⁡(x)+x−xcos⁡(x)x^{- x} \left(- \log{\left(x \right)} - 1\right) \sin{\left(x \right)} + x^{- x} \cos{\left(x \right)}
Simplificar
Segunda derivada [src] 
 -x /          /            2   1\                               \
x  *|-sin(x) + |(1 + log(x))  - -|*sin(x) - 2*(1 + log(x))*cos(x)|
    \          \                x/                               /
x−x(((log⁡(x)+1)2−1x)sin⁡(x)−2(log⁡(x)+1)cos⁡(x)−sin⁡(x))x^{- x} \left(\left(\left(\log{\left(x \right)} + 1\right)^{2} - \frac{1}{x}\right) \sin{\left(x \right)} - 2 \left(\log{\left(x \right)} + 1\right) \cos{\left(x \right)} - \sin{\left(x \right)}\right)
Simplificar
Tercera derivada [src] 
 -x /          /1                3   3*(1 + log(x))\                                    /            2   1\       \
x  *|-cos(x) + |-- - (1 + log(x))  + --------------|*sin(x) + 3*(1 + log(x))*sin(x) + 3*|(1 + log(x))  - -|*cos(x)|
    |          | 2                         x       |                                    \                x/       |
    \          \x                                  /                                                              /
x−x(3((log⁡(x)+1)2−1x)cos⁡(x)+3(log⁡(x)+1)sin⁡(x)+(−(log⁡(x)+1)3+3(log⁡(x)+1)x+1x2)sin⁡(x)−cos⁡(x))x^{- x} \left(3 \left(\left(\log{\left(x \right)} + 1\right)^{2} - \frac{1}{x}\right) \cos{\left(x \right)} + 3 \left(\log{\left(x \right)} + 1\right) \sin{\left(x \right)} + \left(- \left(\log{\left(x \right)} + 1\right)^{3} + \frac{3 \left(\log{\left(x \right)} + 1\right)}{x} + \frac{1}{x^{2}}\right) \sin{\left(x \right)} - \cos{\left(x \right)}\right)
Simplificar
Gráfico
Derivada de x^-xsinx
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