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Derivada de la función/ x^x-1/ x^(x^x-1)*x^x

Derivada de x^(x^x-1)*x^x

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  - 1  x
x      *x
xxxxx−1x^{x} x^{x^{x} - 1} 
x^(x^x - 1)*x^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)=xxx−1f{\left(x \right)} = x^{x^{x} - 1}; 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

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

    g(x)=xxg{\left(x \right)} = x^{x}; calculamos 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)

    Como resultado de: xxxxx−1(log⁡(x)+1)+xx(xx−1)xx−1(log⁡(xx−1)+1)x^{x} x^{x^{x} - 1} \left(\log{\left(x \right)} + 1\right) + x^{x} \left(x^{x} - 1\right)^{x^{x} - 1} \left(\log{\left(x^{x} - 1 \right)} + 1\right)

  2. Simplificamos:

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

Respuesta:

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

Primera derivada [src] 
     x                         x     / x                             \
 x  x  - 1                 x  x  - 1 |x  - 1    x                    |
x *x      *(1 + log(x)) + x *x      *|------ + x *(1 + log(x))*log(x)|
                                     \  x                            /
xxxxx−1(xx(log⁡(x)+1)log⁡(x)+xx−1x)+xxxxx−1(log⁡(x)+1)x^{x} x^{x^{x} - 1} \left(x^{x} \left(\log{\left(x \right)} + 1\right) \log{\left(x \right)} + \frac{x^{x} - 1}{x}\right) + x^{x} x^{x^{x} - 1} \left(\log{\left(x \right)} + 1\right)
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Segunda derivada [src] 
            /                                                      2                                                                                                                        \
          x |                    /      x                         \          x                  /      x                         \    x                                       x             |
 x  -1 + x  |1               2   |-1 + x     x                    |    -1 + x                   |-1 + x     x                    |   x *log(x)    x             2          2*x *(1 + log(x))|
x *x       *|- + (1 + log(x))  + |------- + x *(1 + log(x))*log(x)|  - ------- + 2*(1 + log(x))*|------- + x *(1 + log(x))*log(x)| + --------- + x *(1 + log(x)) *log(x) + -----------------|
            |x                   \   x                            /        2                    \   x                            /       x                                         x        |
            \                                                             x                                                                                                                 /
xxxxx−1(xx(log⁡(x)+1)2log⁡(x)+(xx(log⁡(x)+1)log⁡(x)+xx−1x)2+2(xx(log⁡(x)+1)log⁡(x)+xx−1x)(log⁡(x)+1)+(log⁡(x)+1)2+2xx(log⁡(x)+1)x+xxlog⁡(x)x+1x−xx−1x2)x^{x} x^{x^{x} - 1} \left(x^{x} \left(\log{\left(x \right)} + 1\right)^{2} \log{\left(x \right)} + \left(x^{x} \left(\log{\left(x \right)} + 1\right) \log{\left(x \right)} + \frac{x^{x} - 1}{x}\right)^{2} + 2 \left(x^{x} \left(\log{\left(x \right)} + 1\right) \log{\left(x \right)} + \frac{x^{x} - 1}{x}\right) \left(\log{\left(x \right)} + 1\right) + \left(\log{\left(x \right)} + 1\right)^{2} + \frac{2 x^{x} \left(\log{\left(x \right)} + 1\right)}{x} + \frac{x^{x} \log{\left(x \right)}}{x} + \frac{1}{x} - \frac{x^{x} - 1}{x^{2}}\right)
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Tercera derivada [src] 
            /                                                  3                                                             /                                  2                                                                    \                                                                                                                                                                                                                                                                                  \
          x |                /      x                         \           /      x\                       x                  |/      x                         \          x    x                                       x             |                         /      x                         \     /      x                         \ /        x    x                                       x             \                              x             x                   x             2      x                    |
 x  -1 + x  |            3   |-1 + x     x                    |    1    2*\-1 + x /   3*(1 + log(x))   3*x                   ||-1 + x     x                    |    -1 + x    x *log(x)    x             2          2*x *(1 + log(x))|     /1               2\ |-1 + x     x                    |     |-1 + x     x                    | |  -1 + x    x *log(x)    x             2          2*x *(1 + log(x))|    x             3          x *log(x)   3*x *(1 + log(x))   3*x *(1 + log(x))    3*x *(1 + log(x))*log(x)|
x *x       *|(1 + log(x))  + |------- + x *(1 + log(x))*log(x)|  - -- + ----------- + -------------- + ---- + 3*(1 + log(x))*||------- + x *(1 + log(x))*log(x)|  - ------- + --------- + x *(1 + log(x)) *log(x) + -----------------| + 3*|- + (1 + log(x)) |*|------- + x *(1 + log(x))*log(x)| + 3*|------- + x *(1 + log(x))*log(x)|*|- ------- + --------- + x *(1 + log(x)) *log(x) + -----------------| + x *(1 + log(x)) *log(x) - --------- - ----------------- + ------------------ + ------------------------|
            |                \   x                            /     2         3             x            2                   |\   x                            /        2         x                                         x        |     \x                / \   x                            /     \   x                            / |      2         x                                         x        |                                  2               2                  x                       x            |
            \                                                      x         x                          x                    \                                         x                                                             /                                                                                                   \     x                                                             /                                 x               x                                                        /
xxxxx−1(xx(log⁡(x)+1)3log⁡(x)+(xx(log⁡(x)+1)log⁡(x)+xx−1x)3+3(xx(log⁡(x)+1)log⁡(x)+xx−1x)((log⁡(x)+1)2+1x)+3(xx(log⁡(x)+1)log⁡(x)+xx−1x)(xx(log⁡(x)+1)2log⁡(x)+2xx(log⁡(x)+1)x+xxlog⁡(x)x−xx−1x2)+(log⁡(x)+1)3+3(log⁡(x)+1)(xx(log⁡(x)+1)2log⁡(x)+(xx(log⁡(x)+1)log⁡(x)+xx−1x)2+2xx(log⁡(x)+1)x+xxlog⁡(x)x−xx−1x2)+3xx(log⁡(x)+1)2x+3xx(log⁡(x)+1)log⁡(x)x+3(log⁡(x)+1)x−3xx(log⁡(x)+1)x2−xxlog⁡(x)x2+3xxx2−1x2+2(xx−1)x3)x^{x} x^{x^{x} - 1} \left(x^{x} \left(\log{\left(x \right)} + 1\right)^{3} \log{\left(x \right)} + \left(x^{x} \left(\log{\left(x \right)} + 1\right) \log{\left(x \right)} + \frac{x^{x} - 1}{x}\right)^{3} + 3 \left(x^{x} \left(\log{\left(x \right)} + 1\right) \log{\left(x \right)} + \frac{x^{x} - 1}{x}\right) \left(\left(\log{\left(x \right)} + 1\right)^{2} + \frac{1}{x}\right) + 3 \left(x^{x} \left(\log{\left(x \right)} + 1\right) \log{\left(x \right)} + \frac{x^{x} - 1}{x}\right) \left(x^{x} \left(\log{\left(x \right)} + 1\right)^{2} \log{\left(x \right)} + \frac{2 x^{x} \left(\log{\left(x \right)} + 1\right)}{x} + \frac{x^{x} \log{\left(x \right)}}{x} - \frac{x^{x} - 1}{x^{2}}\right) + \left(\log{\left(x \right)} + 1\right)^{3} + 3 \left(\log{\left(x \right)} + 1\right) \left(x^{x} \left(\log{\left(x \right)} + 1\right)^{2} \log{\left(x \right)} + \left(x^{x} \left(\log{\left(x \right)} + 1\right) \log{\left(x \right)} + \frac{x^{x} - 1}{x}\right)^{2} + \frac{2 x^{x} \left(\log{\left(x \right)} + 1\right)}{x} + \frac{x^{x} \log{\left(x \right)}}{x} - \frac{x^{x} - 1}{x^{2}}\right) + \frac{3 x^{x} \left(\log{\left(x \right)} + 1\right)^{2}}{x} + \frac{3 x^{x} \left(\log{\left(x \right)} + 1\right) \log{\left(x \right)}}{x} + \frac{3 \left(\log{\left(x \right)} + 1\right)}{x} - \frac{3 x^{x} \left(\log{\left(x \right)} + 1\right)}{x^{2}} - \frac{x^{x} \log{\left(x \right)}}{x^{2}} + \frac{3 x^{x}}{x^{2}} - \frac{1}{x^{2}} + \frac{2 \left(x^{x} - 1\right)}{x^{3}}\right)
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