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

Derivada de x^(x^x-1)

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

Ha introducido [src] 
  x    
 x  - 1
x
xxx−1x^{x^{x} - 1} 
x^(x^x - 1)
Solución detallada

  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)

  2. Simplificamos:

    (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)

Respuesta:

(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)

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