Derivada de (x^x)+x^(x^(x))

Solución

Ha introducido [src]

      / x\
 x    \x /
x  + x

$$x^{x} + x^{x^{x}}$$

x^x + x^(x^x)

Solución detallada

diferenciamos $x^{x} + x^{x^{x}}$ miembro por miembro:
1. No logro encontrar los pasos en la búsqueda de esta derivada.
  
  Perola derivada
  
  $x^{x} \left(\log{\left(x \right)} + 1\right)$
2. No logro encontrar los pasos en la búsqueda de esta derivada.
  
  Perola derivada
  
  $\left(\log{\left(x^{x} \right)} + 1\right) \left(x^{x}\right)^{x^{x}}$
Como resultado de: $x^{x} \left(\log{\left(x \right)} + 1\right) + \left(\log{\left(x^{x} \right)} + 1\right) \left(x^{x}\right)^{x^{x}}$

Respuesta:

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

Primera derivada [src]

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

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

Simplificar

Segunda derivada [src]

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

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

Simplificar

Tercera derivada [src]

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

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

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