Derivada de x^x*(ln(1+x^x))/(1+x^x)

Ha introducido [src]

 x    /     x\
x *log\1 + x /
--------------
         x    
    1 + x

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

(x^x*log(1 + x^x))/(1 + x^x)

Solución detallada

Se aplica la regla de la derivada parcial:

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

$f{\left(x \right)} = x^{x} \log{\left(x^{x} + 1 \right)}$ y $g{\left(x \right)} = x^{x} + 1$ .

Para calcular $\frac{d}{d x} f{\left(x \right)}$ :
1. Se aplica la regla de la derivada de una multiplicación:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = x^{x}$ ; calculamos $\frac{d}{d x} f{\left(x \right)}$ :
  1. No logro encontrar los pasos en la búsqueda de esta derivada.
    
    Perola derivada
    
    $x^{x} \left(\log{\left(x \right)} + 1\right)$
  $g{\left(x \right)} = \log{\left(x^{x} + 1 \right)}$ ; calculamos $\frac{d}{d x} g{\left(x \right)}$ :
  1. Sustituimos $u = x^{x} + 1$ .
  2. Derivado $\log{\left(u \right)}$ es $\frac{1}{u}$ .
  3. Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \left(x^{x} + 1\right)$ :
    1. diferenciamos $x^{x} + 1$ miembro por miembro:
      1. La derivada de una constante $1$ es igual a cero.
      2. No logro encontrar los pasos en la búsqueda de esta derivada.
        
        Perola derivada
        
        $x^{x} \left(\log{\left(x \right)} + 1\right)$
      Como resultado de: $x^{x} \left(\log{\left(x \right)} + 1\right)$
    Como resultado de la secuencia de reglas:
    
    $\frac{x^{x} \left(\log{\left(x \right)} + 1\right)}{x^{x} + 1}$
  Como resultado de: $\frac{x^{2 x} \left(\log{\left(x \right)} + 1\right)}{x^{x} + 1} + x^{x} \left(\log{\left(x \right)} + 1\right) \log{\left(x^{x} + 1 \right)}$
Para calcular $\frac{d}{d x} g{\left(x \right)}$ :
1. diferenciamos $x^{x} + 1$ miembro por miembro:
  1. La derivada de una constante $1$ es igual a cero.
  2. No logro encontrar los pasos en la búsqueda de esta derivada.
    
    Perola derivada
    
    $x^{x} \left(\log{\left(x \right)} + 1\right)$
  Como resultado de: $x^{x} \left(\log{\left(x \right)} + 1\right)$
Ahora aplicamos la regla de la derivada de una divesión:

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

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

Respuesta:

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

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

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

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

Simplificar

Segunda derivada [src]

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

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

Simplificar

Tercera derivada [src]

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

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

Simplificar

4-я производная [src]

   /                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          /                                                       /                                         x             3      2*x             3      x             \                                                                                                   \                                                                                                                                                                                                                                                                                                                                                                                                       \
   |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   /                                     /                     x             2\                     \                     |                                                     x |            3   1    3*(1 + log(x))   3*x *(1 + log(x))    2*x   *(1 + log(x))    3*x *(1 + log(x))|                                                             /                     x             2\|                                                                                                                                                                                                                                                                                                                                                                                                       |
   |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |                                   x |1               2   x *(1 + log(x)) |                     |                     |                                                    x *|(1 + log(x))  - -- + -------------- - ------------------ + -------------------- - -----------------|                                              x              |1               2   x *(1 + log(x)) ||                                                                                                                                                                                                                                                                                                                                                                                                       |
   |                                                                              /                                                         2      x             4      3*x             4          x          2*x             4       x             2      x                    2*x             2\      /                                                         2       3*x             4       x             4          x          2*x             4       x             2      x                    2*x             2\                                                             |                                  x *|- + (1 + log(x))  - ----------------|                     |                     |                                                       |                 2         x                     x                      2               /     x\   |      x              /1               2\   3*x *(1 + log(x))*|- + (1 + log(x))  - ----------------||                                                                               /                                         x             3      2*x             3      x             \                                                                                       /   x                \ /                                         x             3      2*x             3      x             \|
   |                                                                            x |            4   2    3    4*(1 + log(x))   6*(1 + log(x))    7*x *(1 + log(x))    6*x   *(1 + log(x))        3*x       12*x   *(1 + log(x))    18*x *(1 + log(x))    4*x *(1 + log(x))   12*x   *(1 + log(x)) |    x |            4   2    3    4*(1 + log(x))   6*(1 + log(x))    24*x   *(1 + log(x))    14*x *(1 + log(x))        6*x       36*x   *(1 + log(x))    36*x *(1 + log(x))    8*x *(1 + log(x))   36*x   *(1 + log(x)) |    /     x\        /                       x             2\ |                                     |x                             x     |      x             2|                     |                                                       |                x                           1 + x               /     x\              x*\1 + x /   |   3*x *(1 + log(x))*|- + (1 + log(x)) |                     |x                             x     ||                                                                x              |            3   1    3*(1 + log(x))   3*x *(1 + log(x))    2*x   *(1 + log(x))    3*x *(1 + log(x))|                            /                     x             2\      x              |  x         /     x\| |            3   1    3*(1 + log(x))   6*x *(1 + log(x))    6*x   *(1 + log(x))    6*x *(1 + log(x))||
   |                                                                           x *|(1 + log(x))  + -- + -- - -------------- + --------------- - ------------------ - -------------------- - ----------- + --------------------- - ------------------- + ----------------- + ---------------------|   x *|(1 + log(x))  + -- + -- - -------------- + --------------- - --------------------- - ------------------- - ----------- + --------------------- - ------------------- + ----------------- + ---------------------|*log\1 + x /      x |1               2   2*x *(1 + log(x)) | |/1               2\    /     x\      \                         1 + x      /   2*x *(1 + log(x)) |      x              |/            3   1    3*(1 + log(x))\    /     x\      \                                                                \1 + x /                           /                     \x                /                     \                         1 + x      /|      x              /            3   1    3*(1 + log(x))\   4*x *(1 + log(x))*|(1 + log(x))  - -- + -------------- - ------------------ + -------------------- - -----------------|      x /1               2\ |1               2   x *(1 + log(x)) |   4*x *(1 + log(x))*|------ + log\1 + x /|*|(1 + log(x))  - -- + -------------- - ------------------ + -------------------- - -----------------||
   |                                                                              |                 3    2          2                x                     x                      3          2 /     x\                 2                /     x\           2 /     x\                     2     |      |                 3    2          2                x                        3                     x          2 /     x\                 2                /     x\           2 /     x\                     2     |               6*x *|- + (1 + log(x))  - ------------------|*||- + (1 + log(x)) |*log\1 + x / + ----------------------------------------- + ------------------|   4*x *(1 + log(x))*||(1 + log(x))  - -- + --------------|*log\1 + x / + -------------------------------------------------------------------------------------------------------- + ------------------------------------- + --------------------------------------------------------|   4*x *(1 + log(x))*|(1 + log(x))  - -- + --------------|                     |                 2         x                     x                      2               /     x\   |   6*x *|- + (1 + log(x)) |*|- + (1 + log(x))  - ----------------|                     |     x              | |                 2         x                     x                      2               /     x\   ||
   |/                                                         2\                  |                x    x          x                                  1 + x               /     x\          x *\1 + x /         /     x\               x*\1 + x /          x *\1 + x /             /     x\      |      |                x    x          x                                  /     x\                 1 + x          x *\1 + x /         /     x\               x*\1 + x /          x *\1 + x /             /     x\      |                    |x                              x      | |\x                /                                      x                               x      |                     ||                 2         x       |                                                                     x                                                                         x                                                x                         |                     |                 2         x       |                     |                x                           1 + x               /     x\              x*\1 + x /   |        \x                / |x                             x     |                     \1 + x               / |                x                           1 + x               /     x\              x*\1 + x /   ||
 x ||            4   2    3    4*(1 + log(x))   6*(1 + log(x)) |    /     x\      \                                                                                       \1 + x /                              \1 + x /                                                         x*\1 + x /      /      \                                                                   \1 + x /                                                    \1 + x /                                                         x*\1 + x /      /                    \                          1 + x       / \                                                    1 + x                           1 + x       /                     \\                x                  /                                                                1 + x                                                                     1 + x                                            1 + x                          /                     \                x                  /                     \                                                                \1 + x /                           /                            \                         1 + x      /                                            \                                                                \1 + x /                           /|
x *||(1 + log(x))  + -- + -- - -------------- + ---------------|*log\1 + x / + ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------------ - ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------- + --------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------------------|
   ||                 3    2          2                x       |                                                                                                                           x                                                                                                                                                                                                                            x                                                                                                                                                                                          x                                                                                                                                                                                                                    x                                                                                                                                                                       x                                                                                         x                                                                                             x                                                                                                        x                                                                    |
   \\                x    x          x                         /                                                                                                                      1 + x                                                                                                                                                                                                                        1 + x                                                                                                                                                                                      1 + x                                                                                                                                                                                                                1 + x                                                                                                                                                                   1 + x                                                                                     1 + x                                                                                         1 + x                                                                                                    1 + x                                                                     /
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\frac{x^{x} \left(- \frac{4 x^{x} \left(\frac{x^{x}}{x^{x} + 1} + \log{\left(x^{x} + 1 \right)}\right) \left(\log{\left(x \right)} + 1\right) \left(\frac{6 x^{2 x} \left(\log{\left(x \right)} + 1\right)^{3}}{\left(x^{x} + 1\right)^{2}} - \frac{6 x^{x} \left(\log{\left(x \right)} + 1\right)^{3}}{x^{x} + 1} + \left(\log{\left(x \right)} + 1\right)^{3} - \frac{6 x^{x} \left(\log{\left(x \right)} + 1\right)}{x \left(x^{x} + 1\right)} + \frac{3 \left(\log{\left(x \right)} + 1\right)}{x} - \frac{1}{x^{2}}\right)}{x^{x} + 1} + \frac{6 x^{x} \left(\left(\log{\left(x \right)} + 1\right)^{2} + \frac{1}{x}\right) \left(- \frac{x^{x} \left(\log{\left(x \right)} + 1\right)^{2}}{x^{x} + 1} + \left(\log{\left(x \right)} + 1\right)^{2} + \frac{1}{x}\right)}{x^{x} + 1} + \frac{4 x^{x} \left(\log{\left(x \right)} + 1\right) \left(\left(\log{\left(x \right)} + 1\right)^{3} + \frac{3 \left(\log{\left(x \right)} + 1\right)}{x} - \frac{1}{x^{2}}\right)}{x^{x} + 1} - \frac{4 x^{x} \left(\log{\left(x \right)} + 1\right) \left(\frac{3 x^{x} \left(\left(\log{\left(x \right)} + 1\right)^{2} + \frac{1}{x}\right) \left(\log{\left(x \right)} + 1\right)}{x^{x} + 1} + \frac{3 x^{x} \left(\log{\left(x \right)} + 1\right) \left(- \frac{x^{x} \left(\log{\left(x \right)} + 1\right)^{2}}{x^{x} + 1} + \left(\log{\left(x \right)} + 1\right)^{2} + \frac{1}{x}\right)}{x^{x} + 1} + \frac{x^{x} \left(\frac{2 x^{2 x} \left(\log{\left(x \right)} + 1\right)^{3}}{\left(x^{x} + 1\right)^{2}} - \frac{3 x^{x} \left(\log{\left(x \right)} + 1\right)^{3}}{x^{x} + 1} + \left(\log{\left(x \right)} + 1\right)^{3} - \frac{3 x^{x} \left(\log{\left(x \right)} + 1\right)}{x \left(x^{x} + 1\right)} + \frac{3 \left(\log{\left(x \right)} + 1\right)}{x} - \frac{1}{x^{2}}\right)}{x^{x} + 1} + \left(\left(\log{\left(x \right)} + 1\right)^{3} + \frac{3 \left(\log{\left(x \right)} + 1\right)}{x} - \frac{1}{x^{2}}\right) \log{\left(x^{x} + 1 \right)}\right)}{x^{x} + 1} + \frac{4 x^{x} \left(\log{\left(x \right)} + 1\right) \left(\frac{2 x^{2 x} \left(\log{\left(x \right)} + 1\right)^{3}}{\left(x^{x} + 1\right)^{2}} - \frac{3 x^{x} \left(\log{\left(x \right)} + 1\right)^{3}}{x^{x} + 1} + \left(\log{\left(x \right)} + 1\right)^{3} - \frac{3 x^{x} \left(\log{\left(x \right)} + 1\right)}{x \left(x^{x} + 1\right)} + \frac{3 \left(\log{\left(x \right)} + 1\right)}{x} - \frac{1}{x^{2}}\right)}{x^{x} + 1} - \frac{6 x^{x} \left(- \frac{2 x^{x} \left(\log{\left(x \right)} + 1\right)^{2}}{x^{x} + 1} + \left(\log{\left(x \right)} + 1\right)^{2} + \frac{1}{x}\right) \left(\frac{2 x^{x} \left(\log{\left(x \right)} + 1\right)^{2}}{x^{x} + 1} + \frac{x^{x} \left(- \frac{x^{x} \left(\log{\left(x \right)} + 1\right)^{2}}{x^{x} + 1} + \left(\log{\left(x \right)} + 1\right)^{2} + \frac{1}{x}\right)}{x^{x} + 1} + \left(\left(\log{\left(x \right)} + 1\right)^{2} + \frac{1}{x}\right) \log{\left(x^{x} + 1 \right)}\right)}{x^{x} + 1} - \frac{x^{x} \left(- \frac{24 x^{3 x} \left(\log{\left(x \right)} + 1\right)^{4}}{\left(x^{x} + 1\right)^{3}} + \frac{36 x^{2 x} \left(\log{\left(x \right)} + 1\right)^{4}}{\left(x^{x} + 1\right)^{2}} - \frac{14 x^{x} \left(\log{\left(x \right)} + 1\right)^{4}}{x^{x} + 1} + \left(\log{\left(x \right)} + 1\right)^{4} + \frac{36 x^{2 x} \left(\log{\left(x \right)} + 1\right)^{2}}{x \left(x^{x} + 1\right)^{2}} - \frac{36 x^{x} \left(\log{\left(x \right)} + 1\right)^{2}}{x \left(x^{x} + 1\right)} + \frac{6 \left(\log{\left(x \right)} + 1\right)^{2}}{x} + \frac{8 x^{x} \left(\log{\left(x \right)} + 1\right)}{x^{2} \left(x^{x} + 1\right)} - \frac{6 x^{x}}{x^{2} \left(x^{x} + 1\right)} - \frac{4 \left(\log{\left(x \right)} + 1\right)}{x^{2}} + \frac{3}{x^{2}} + \frac{2}{x^{3}}\right) \log{\left(x^{x} + 1 \right)}}{x^{x} + 1} + \frac{x^{x} \left(- \frac{6 x^{3 x} \left(\log{\left(x \right)} + 1\right)^{4}}{\left(x^{x} + 1\right)^{3}} + \frac{12 x^{2 x} \left(\log{\left(x \right)} + 1\right)^{4}}{\left(x^{x} + 1\right)^{2}} - \frac{7 x^{x} \left(\log{\left(x \right)} + 1\right)^{4}}{x^{x} + 1} + \left(\log{\left(x \right)} + 1\right)^{4} + \frac{12 x^{2 x} \left(\log{\left(x \right)} + 1\right)^{2}}{x \left(x^{x} + 1\right)^{2}} - \frac{18 x^{x} \left(\log{\left(x \right)} + 1\right)^{2}}{x \left(x^{x} + 1\right)} + \frac{6 \left(\log{\left(x \right)} + 1\right)^{2}}{x} + \frac{4 x^{x} \left(\log{\left(x \right)} + 1\right)}{x^{2} \left(x^{x} + 1\right)} - \frac{3 x^{x}}{x^{2} \left(x^{x} + 1\right)} - \frac{4 \left(\log{\left(x \right)} + 1\right)}{x^{2}} + \frac{3}{x^{2}} + \frac{2}{x^{3}}\right)}{x^{x} + 1} + \left(\left(\log{\left(x \right)} + 1\right)^{4} + \frac{6 \left(\log{\left(x \right)} + 1\right)^{2}}{x} - \frac{4 \left(\log{\left(x \right)} + 1\right)}{x^{2}} + \frac{3}{x^{2}} + \frac{2}{x^{3}}\right) \log{\left(x^{x} + 1 \right)}\right)}{x^{x} + 1}

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Derivada de x^x*(ln(1+x^x))/(1+x^x)

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