Derivada de x^(x^x)/x

Solución

Ha introducido [src]

 / x\
 \x /
x    
-----
  x

$$\frac{x^{x^{x}}}{x}$$

x^(x^x)/x

Solución detallada

Se aplica la regla de la derivada parcial:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = x^{x^{x}}$ y $g{\left(x \right)} = x$ .

Para calcular $\frac{d}{d x} f{\left(x \right)}$ :
1. No logro encontrar los pasos en la búsqueda de esta derivada.
  
  Perola derivada
  
  $\left(\log{\left(x^{x} \right)} + 1\right) \left(x^{x}\right)^{x^{x}}$
Para calcular $\frac{d}{d x} g{\left(x \right)}$ :
1. Según el principio, aplicamos: $x$ tenemos $1$
Ahora aplicamos la regla de la derivada de una divesión:

$\frac{x \left(\log{\left(x^{x} \right)} + 1\right) \left(x^{x}\right)^{x^{x}} - x^{x^{x}}}{x^{2}}$

Respuesta:

$\frac{x \left(\log{\left(x^{x} \right)} + 1\right) \left(x^{x}\right)^{x^{x}} - x^{x^{x}}}{x^{2}}$

Primera derivada [src]

           / x\ / x                         \
   / x\    \x / |x     x                    |
   \x /   x    *|-- + x *(1 + log(x))*log(x)|
  x             \x                          /
- ----- + -----------------------------------
     2                     x                 
    x

$$\frac{x^{x^{x}} \left(x^{x} \left(\log{\left(x \right)} + 1\right) \log{\left(x \right)} + \frac{x^{x}}{x}\right)}{x} - \frac{x^{x^{x}}}{x^{2}}$$

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Segunda derivada [src]

      /                                                                                                     x /1                      \\
 / x\ |        /                                            2                                        \   2*x *|- + (1 + log(x))*log(x)||
 \x / |2     x |  1    log(x)    x /1                      \                2          2*(1 + log(x))|        \x                      /|
x    *|-- + x *|- -- + ------ + x *|- + (1 + log(x))*log(x)|  + (1 + log(x)) *log(x) + --------------| - ------------------------------|
      | 2      |   2     x         \x                      /                                 x       |                 x               |
      \x       \  x                                                                                  /                                 /
----------------------------------------------------------------------------------------------------------------------------------------
                                                                   x

$$\frac{x^{x^{x}} \left(x^{x} \left(x^{x} \left(\left(\log{\left(x \right)} + 1\right) \log{\left(x \right)} + \frac{1}{x}\right)^{2} + \left(\log{\left(x \right)} + 1\right)^{2} \log{\left(x \right)} + \frac{2 \left(\log{\left(x \right)} + 1\right)}{x} + \frac{\log{\left(x \right)}}{x} - \frac{1}{x^{2}}\right) - \frac{2 x^{x} \left(\left(\log{\left(x \right)} + 1\right) \log{\left(x \right)} + \frac{1}{x}\right)}{x} + \frac{2}{x^{2}}\right)}{x}$$

Simplificar

Tercera derivada [src]

      /                                                                                                                                                                                                                                                 /                                            2                                        \                                 \
      |                                                                                                                                                                                                                                               x |  1    log(x)    x /1                      \                2          2*(1 + log(x))|                                 |
      |                                                                                                                                                                                                                                            3*x *|- -- + ------ + x *|- + (1 + log(x))*log(x)|  + (1 + log(x)) *log(x) + --------------|      x /1                      \|
 / x\ |          /                                        3                                                                  2                                                                                                                 \        |   2     x         \x                      /                                 x       |   6*x *|- + (1 + log(x))*log(x)||
 \x / |  6     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 *|-- + -- + x   *|- + (1 + log(x))*log(x)|  + (1 + log(x)) *log(x) - ------ - -------------- + --------------- + --------------------- + 3*x *|- + (1 + log(x))*log(x)|*|- -- + ------ + (1 + log(x)) *log(x) + --------------|| - -------------------------------------------------------------------------------------------- + ------------------------------|
      |   3      | 3    2        \x                      /                              2            2                x                    x                  \x                      / |   2     x                                   x       ||                                                x                                                                2              |
      \  x       \x    x                                                               x            x                                                                                   \  x                                                  //                                                                                                                x               /
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                                                                        x

$$\frac{x^{x^{x}} \left(x^{x} \left(x^{2 x} \left(\left(\log{\left(x \right)} + 1\right) \log{\left(x \right)} + \frac{1}{x}\right)^{3} + 3 x^{x} \left(\left(\log{\left(x \right)} + 1\right) \log{\left(x \right)} + \frac{1}{x}\right) \left(\left(\log{\left(x \right)} + 1\right)^{2} \log{\left(x \right)} + \frac{2 \left(\log{\left(x \right)} + 1\right)}{x} + \frac{\log{\left(x \right)}}{x} - \frac{1}{x^{2}}\right) + \left(\log{\left(x \right)} + 1\right)^{3} \log{\left(x \right)} + \frac{3 \left(\log{\left(x \right)} + 1\right)^{2}}{x} + \frac{3 \left(\log{\left(x \right)} + 1\right) \log{\left(x \right)}}{x} - \frac{3 \left(\log{\left(x \right)} + 1\right)}{x^{2}} - \frac{\log{\left(x \right)}}{x^{2}} + \frac{3}{x^{2}} + \frac{2}{x^{3}}\right) - \frac{3 x^{x} \left(x^{x} \left(\left(\log{\left(x \right)} + 1\right) \log{\left(x \right)} + \frac{1}{x}\right)^{2} + \left(\log{\left(x \right)} + 1\right)^{2} \log{\left(x \right)} + \frac{2 \left(\log{\left(x \right)} + 1\right)}{x} + \frac{\log{\left(x \right)}}{x} - \frac{1}{x^{2}}\right)}{x} + \frac{6 x^{x} \left(\left(\log{\left(x \right)} + 1\right) \log{\left(x \right)} + \frac{1}{x}\right)}{x^{2}} - \frac{6}{x^{3}}\right)}{x}$$

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