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Derivada de la función/ y=x^2/ xsinx/ x^2^x/ y=x^2^xsinx

Derivada de y=x^2^xsinx

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

Ha introducido [src] 
 / x\       
 \2 /       
x    *sin(x)
x2xsin⁡(x)x^{2^{x}} \sin{\left(x \right)} 
x^(2^x)*sin(x)
Solución detallada

  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)=x2xf{\left(x \right)} = x^{2^{x}}; calculamos ddxf(x)\frac{d}{d x} f{\left(x \right)}:

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

      Perola derivada

      22xx(log⁡(2x)+1)2^{2^{x} x} \left(\log{\left(2^{x} \right)} + 1\right)

    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: 22xx(log⁡(2x)+1)sin⁡(x)+x2xcos⁡(x)2^{2^{x} x} \left(\log{\left(2^{x} \right)} + 1\right) \sin{\left(x \right)} + x^{2^{x}} \cos{\left(x \right)}

  2. Simplificamos:

    22xx(xlog⁡(2)+1)sin⁡(x)+x2xcos⁡(x)2^{2^{x} x} \left(x \log{\left(2 \right)} + 1\right) \sin{\left(x \right)} + x^{2^{x}} \cos{\left(x \right)}

Respuesta:

22xx(xlog⁡(2)+1)sin⁡(x)+x2xcos⁡(x)2^{2^{x} x} \left(x \log{\left(2 \right)} + 1\right) \sin{\left(x \right)} + x^{2^{x}} \cos{\left(x \right)}

Primera derivada [src] 
 / x\           / x\ / x                   \       
 \2 /           \2 / |2     x              |       
x    *cos(x) + x    *|-- + 2 *log(2)*log(x)|*sin(x)
                     \x                    /
x2x(2xlog⁡(2)log⁡(x)+2xx)sin⁡(x)+x2xcos⁡(x)x^{2^{x}} \left(2^{x} \log{\left(2 \right)} \log{\left(x \right)} + \frac{2^{x}}{x}\right) \sin{\left(x \right)} + x^{2^{x}} \cos{\left(x \right)}
Simplificar
Segunda derivada [src] 
 / x\ /             /                             2                            \                                         \
 \2 / |           x |  1     x /1                \       2             2*log(2)|             x /1                \       |
x    *|-sin(x) + 2 *|- -- + 2 *|- + log(2)*log(x)|  + log (2)*log(x) + --------|*sin(x) + 2*2 *|- + log(2)*log(x)|*cos(x)|
      |             |   2      \x                /                        x    |               \x                /       |
      \             \  x                                                       /                                         /
x2x(2⋅2x(log⁡(2)log⁡(x)+1x)cos⁡(x)+2x(2x(log⁡(2)log⁡(x)+1x)2+log⁡(2)2log⁡(x)+2log⁡(2)x−1x2)sin⁡(x)−sin⁡(x))x^{2^{x}} \left(2 \cdot 2^{x} \left(\log{\left(2 \right)} \log{\left(x \right)} + \frac{1}{x}\right) \cos{\left(x \right)} + 2^{x} \left(2^{x} \left(\log{\left(2 \right)} \log{\left(x \right)} + \frac{1}{x}\right)^{2} + \log{\left(2 \right)}^{2} \log{\left(x \right)} + \frac{2 \log{\left(2 \right)}}{x} - \frac{1}{x^{2}}\right) \sin{\left(x \right)} - \sin{\left(x \right)}\right)
Simplificar
Tercera derivada [src] 
 / x\ /             /                             3                                    2                                                                 \                                                 /                             2                            \       \
 \2 / |           x |2     2*x /1                \       3             3*log(2)   3*log (2)      x /1                \ /  1       2             2*log(2)\|             x /1                \             x |  1     x /1                \       2             2*log(2)|       |
x    *|-cos(x) + 2 *|-- + 2   *|- + log(2)*log(x)|  + log (2)*log(x) - -------- + --------- + 3*2 *|- + log(2)*log(x)|*|- -- + log (2)*log(x) + --------||*sin(x) - 3*2 *|- + log(2)*log(x)|*sin(x) + 3*2 *|- -- + 2 *|- + log(2)*log(x)|  + log (2)*log(x) + --------|*cos(x)|
      |             | 3        \x                /                         2          x            \x                / |   2                       x    ||               \x                /               |   2      \x                /                        x    |       |
      \             \x                                                    x                                            \  x                             //                                                 \  x                                                       /       /
x2x(−3⋅2x(log⁡(2)log⁡(x)+1x)sin⁡(x)+3⋅2x(2x(log⁡(2)log⁡(x)+1x)2+log⁡(2)2log⁡(x)+2log⁡(2)x−1x2)cos⁡(x)+2x(22x(log⁡(2)log⁡(x)+1x)3+3⋅2x(log⁡(2)log⁡(x)+1x)(log⁡(2)2log⁡(x)+2log⁡(2)x−1x2)+log⁡(2)3log⁡(x)+3log⁡(2)2x−3log⁡(2)x2+2x3)sin⁡(x)−cos⁡(x))x^{2^{x}} \left(- 3 \cdot 2^{x} \left(\log{\left(2 \right)} \log{\left(x \right)} + \frac{1}{x}\right) \sin{\left(x \right)} + 3 \cdot 2^{x} \left(2^{x} \left(\log{\left(2 \right)} \log{\left(x \right)} + \frac{1}{x}\right)^{2} + \log{\left(2 \right)}^{2} \log{\left(x \right)} + \frac{2 \log{\left(2 \right)}}{x} - \frac{1}{x^{2}}\right) \cos{\left(x \right)} + 2^{x} \left(2^{2 x} \left(\log{\left(2 \right)} \log{\left(x \right)} + \frac{1}{x}\right)^{3} + 3 \cdot 2^{x} \left(\log{\left(2 \right)} \log{\left(x \right)} + \frac{1}{x}\right) \left(\log{\left(2 \right)}^{2} \log{\left(x \right)} + \frac{2 \log{\left(2 \right)}}{x} - \frac{1}{x^{2}}\right) + \log{\left(2 \right)}^{3} \log{\left(x \right)} + \frac{3 \log{\left(2 \right)}^{2}}{x} - \frac{3 \log{\left(2 \right)}}{x^{2}} + \frac{2}{x^{3}}\right) \sin{\left(x \right)} - \cos{\left(x \right)}\right)
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