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

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

Solución

Ha introducido [src]

 / x\       
 \x /  x - 1
x    *x

x^{x^{x}} x^{x - 1}

x^(x^x)*x^(x - 1)

Solución detallada

Se aplica la regla de la derivada de una multiplicación:

$\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$

$f{\left(x \right)} = x^{x^{x}}$ ; calculamos $\frac{d}{d x} f{\left(x \right)}$ :
1. No logro encontrar los pasos en la búsqueda de esta derivada.
  
  Perola derivada
  
  $\left(\log{\left(x^{x} \right)} + 1\right) \left(x^{x}\right)^{x^{x}}$
$g{\left(x \right)} = x^{x - 1}$ ; calculamos $\frac{d}{d x} g{\left(x \right)}$ :
1. No logro encontrar los pasos en la búsqueda de esta derivada.
  
  Perola derivada
  
  $\left(x - 1\right)^{x - 1} \left(\log{\left(x - 1 \right)} + 1\right)$
Como resultado de: $x^{x^{x}} \left(x - 1\right)^{x - 1} \left(\log{\left(x - 1 \right)} + 1\right) + x^{x - 1} \left(\log{\left(x^{x} \right)} + 1\right) \left(x^{x}\right)^{x^{x}}$
Simplificamos:

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

Respuesta:

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

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           /

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)

Simplificar

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                                                                                  /                                                   /

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)

Simplificar

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                                                                                  //

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)

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

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