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

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

Ha introducido [src]
 / x\       
 \x /  x - 1
x    *x     
xxxxx1x^{x^{x}} x^{x - 1}
x^(x^x)*x^(x - 1)
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)=xxxf{\left(x \right)} = x^{x^{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

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

    g(x)=xx1g{\left(x \right)} = x^{x - 1}; 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

      (x1)x1(log(x1)+1)\left(x - 1\right)^{x - 1} \left(\log{\left(x - 1 \right)} + 1\right)

    Como resultado de: xxx(x1)x1(log(x1)+1)+xx1(log(xx)+1)(xx)xxx^{x^{x}} \left(x - 1\right)^{x - 1} \left(\log{\left(x - 1 \right)} + 1\right) + x^{x - 1} \left(\log{\left(x^{x} \right)} + 1\right) \left(x^{x}\right)^{x^{x}}

  2. Simplificamos:

    xxx(x1)x1(log(x1)+1)+xx1(log(xx)+1)(xx)xxx^{x^{x}} \left(x - 1\right)^{x - 1} \left(\log{\left(x - 1 \right)} + 1\right) + x^{x - 1} \left(\log{\left(x^{x} \right)} + 1\right) \left(x^{x}\right)^{x^{x}}


Respuesta:

xxx(x1)x1(log(x1)+1)+xx1(log(xx)+1)(xx)xxx^{x^{x}} \left(x - 1\right)^{x - 1} \left(\log{\left(x - 1 \right)} + 1\right) + x^{x - 1} \left(\log{\left(x^{x} \right)} + 1\right) \left(x^{x}\right)^{x^{x}}

Primera derivada [src]
 / x\        / x                         \    / x\                        
 \x /  x - 1 |x     x                    |    \x /  x - 1 /x - 1         \
x    *x     *|-- + x *(1 + log(x))*log(x)| + x    *x     *|----- + log(x)|
             \x                          /                \  x           /
xxxxx1(xx(log(x)+1)log(x)+xxx)+xxxxx1(log(x)+x1x)x^{x^{x}} x^{x - 1} \left(x^{x} \left(\log{\left(x \right)} + 1\right) \log{\left(x \right)} + \frac{x^{x}}{x}\right) + x^{x^{x}} x^{x - 1} \left(\log{\left(x \right)} + \frac{x - 1}{x}\right)
Segunda derivada [src]
              /                         -1 + x                                                                                                                                                \
 / x\         |                 2   2 - ------      /                                            2                                        \                                                   |
 \x /  -1 + x |/-1 + x         \          x       x |  1    log(x)    x /1                      \                2          2*(1 + log(x))|      x /1                      \ /-1 + x         \|
x    *x      *||------ + log(x)|  + ---------- + x *|- -- + ------ + x *|- + (1 + log(x))*log(x)|  + (1 + log(x)) *log(x) + --------------| + 2*x *|- + (1 + log(x))*log(x)|*|------ + log(x)||
              |\  x            /        x           |   2     x         \x                      /                                 x       |        \x                      / \  x            /|
              \                                     \  x                                                                                  /                                                   /
xxxxx1(2xx((log(x)+1)log(x)+1x)(log(x)+x1x)+xx(xx((log(x)+1)log(x)+1x)2+(log(x)+1)2log(x)+2(log(x)+1)x+log(x)x1x2)+(log(x)+x1x)2+2x1xx)x^{x^{x}} x^{x - 1} \left(2 x^{x} \left(\left(\log{\left(x \right)} + 1\right) \log{\left(x \right)} + \frac{1}{x}\right) \left(\log{\left(x \right)} + \frac{x - 1}{x}\right) + x^{x} \left(x^{x} \left(\left(\log{\left(x \right)} + 1\right) \log{\left(x \right)} + \frac{1}{x}\right)^{2} + \left(\log{\left(x \right)} + 1\right)^{2} \log{\left(x \right)} + \frac{2 \left(\log{\left(x \right)} + 1\right)}{x} + \frac{\log{\left(x \right)}}{x} - \frac{1}{x^{2}}\right) + \left(\log{\left(x \right)} + \frac{x - 1}{x}\right)^{2} + \frac{2 - \frac{x - 1}{x}}{x}\right)
Tercera derivada [src]
              /                                                                                                                                                                                                                                                              2*(-1 + x)     /    -1 + x\ /-1 + x         \                                  /                         -1 + x\                                                                                                                 \
 / x\         |                 3      /                                        3                                                                  2                                                                                                                 \   3 - ----------   3*|2 - ------|*|------ + log(x)|                                  |                 2   2 - ------|                          /                                            2                                        \|
 \x /  -1 + x |/-1 + x         \     x |2    3     2*x /1                      \                3          log(x)   3*(1 + log(x))   3*(1 + log(x))    3*(1 + log(x))*log(x)      x /1                      \ /  1    log(x)               2          2*(1 + log(x))\|           x          \      x   / \  x            /      x /1                      \ |/-1 + x         \          x   |      x /-1 + x         \ |  1    log(x)    x /1                      \                2          2*(1 + log(x))||
x    *x      *||------ + log(x)|  + x *|-- + -- + x   *|- + (1 + log(x))*log(x)|  + (1 + log(x)) *log(x) - ------ - -------------- + --------------- + --------------------- + 3*x *|- + (1 + log(x))*log(x)|*|- -- + ------ + (1 + log(x)) *log(x) + --------------|| - -------------- + -------------------------------- + 3*x *|- + (1 + log(x))*log(x)|*||------ + log(x)|  + ----------| + 3*x *|------ + log(x)|*|- -- + ------ + x *|- + (1 + log(x))*log(x)|  + (1 + log(x)) *log(x) + --------------||
              |\  x            /       | 3    2        \x                      /                              2            2                x                    x                  \x                      / |   2     x                                   x       ||          2                        x                        \x                      / \\  x            /        x     /        \  x            / |   2     x         \x                      /                                 x       ||
              \                        \x    x                                                               x            x                                                                                   \  x                                                  //         x                                                                                                                                       \  x                                                                                  //
xxxxx1(3xx((log(x)+1)log(x)+1x)((log(x)+x1x)2+2x1xx)+3xx(log(x)+x1x)(xx((log(x)+1)log(x)+1x)2+(log(x)+1)2log(x)+2(log(x)+1)x+log(x)x1x2)+xx(x2x((log(x)+1)log(x)+1x)3+3xx((log(x)+1)log(x)+1x)((log(x)+1)2log(x)+2(log(x)+1)x+log(x)x1x2)+(log(x)+1)3log(x)+3(log(x)+1)2x+3(log(x)+1)log(x)x3(log(x)+1)x2log(x)x2+3x2+2x3)+(log(x)+x1x)3+3(2x1x)(log(x)+x1x)x32(x1)xx2)x^{x^{x}} x^{x - 1} \left(3 x^{x} \left(\left(\log{\left(x \right)} + 1\right) \log{\left(x \right)} + \frac{1}{x}\right) \left(\left(\log{\left(x \right)} + \frac{x - 1}{x}\right)^{2} + \frac{2 - \frac{x - 1}{x}}{x}\right) + 3 x^{x} \left(\log{\left(x \right)} + \frac{x - 1}{x}\right) \left(x^{x} \left(\left(\log{\left(x \right)} + 1\right) \log{\left(x \right)} + \frac{1}{x}\right)^{2} + \left(\log{\left(x \right)} + 1\right)^{2} \log{\left(x \right)} + \frac{2 \left(\log{\left(x \right)} + 1\right)}{x} + \frac{\log{\left(x \right)}}{x} - \frac{1}{x^{2}}\right) + x^{x} \left(x^{2 x} \left(\left(\log{\left(x \right)} + 1\right) \log{\left(x \right)} + \frac{1}{x}\right)^{3} + 3 x^{x} \left(\left(\log{\left(x \right)} + 1\right) \log{\left(x \right)} + \frac{1}{x}\right) \left(\left(\log{\left(x \right)} + 1\right)^{2} \log{\left(x \right)} + \frac{2 \left(\log{\left(x \right)} + 1\right)}{x} + \frac{\log{\left(x \right)}}{x} - \frac{1}{x^{2}}\right) + \left(\log{\left(x \right)} + 1\right)^{3} \log{\left(x \right)} + \frac{3 \left(\log{\left(x \right)} + 1\right)^{2}}{x} + \frac{3 \left(\log{\left(x \right)} + 1\right) \log{\left(x \right)}}{x} - \frac{3 \left(\log{\left(x \right)} + 1\right)}{x^{2}} - \frac{\log{\left(x \right)}}{x^{2}} + \frac{3}{x^{2}} + \frac{2}{x^{3}}\right) + \left(\log{\left(x \right)} + \frac{x - 1}{x}\right)^{3} + \frac{3 \left(2 - \frac{x - 1}{x}\right) \left(\log{\left(x \right)} + \frac{x - 1}{x}\right)}{x} - \frac{3 - \frac{2 \left(x - 1\right)}{x}}{x^{2}}\right)