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Derivada de la función/ lnx*x/ x^(x)/ x^(x^(x))lnx*x^(x)lnx

Derivada de x^(x^(x))lnx*x^(x)lnx

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

Ha introducido [src] 
 / x\                 
 \x /         x       
x    *log(x)*x *log(x)
xxxxxlog⁡(x)log⁡(x)x^{x} x^{x^{x}} \log{\left(x \right)} \log{\left(x \right)} 
((x^(x^x)*log(x))*x^x)*log(x)
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)=xxxxxlog⁡(x)f{\left(x \right)} = x^{x} x^{x^{x}} \log{\left(x \right)}; calculamos ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    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)=xxxlog⁡(x)f{\left(x \right)} = x^{x^{x}} \log{\left(x \right)}; calculamos ddxf(x)\frac{d}{d x} f{\left(x \right)}:

      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)=log⁡(x)g{\left(x \right)} = \log{\left(x \right)}; calculamos ddxg(x)\frac{d}{d x} g{\left(x \right)}:

        1. Derivado log⁡(x)\log{\left(x \right)} es 1x\frac{1}{x}.

        Como resultado de: (log⁡(xx)+1)(xx)xxlog⁡(x)+xxxx\left(\log{\left(x^{x} \right)} + 1\right) \left(x^{x}\right)^{x^{x}} \log{\left(x \right)} + \frac{x^{x^{x}}}{x}

      g(x)=xxg{\left(x \right)} = x^{x}; 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

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

      Como resultado de: xxxxx(log⁡(x)+1)log⁡(x)+xx((log⁡(xx)+1)(xx)xxlog⁡(x)+xxxx)x^{x} x^{x^{x}} \left(\log{\left(x \right)} + 1\right) \log{\left(x \right)} + x^{x} \left(\left(\log{\left(x^{x} \right)} + 1\right) \left(x^{x}\right)^{x^{x}} \log{\left(x \right)} + \frac{x^{x^{x}}}{x}\right)

    g(x)=log⁡(x)g{\left(x \right)} = \log{\left(x \right)}; calculamos ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. Derivado log⁡(x)\log{\left(x \right)} es 1x\frac{1}{x}.

    Como resultado de: (xxxxx(log⁡(x)+1)log⁡(x)+xx((log⁡(xx)+1)(xx)xxlog⁡(x)+xxxx))log⁡(x)+xxxxxlog⁡(x)x\left(x^{x} x^{x^{x}} \left(\log{\left(x \right)} + 1\right) \log{\left(x \right)} + x^{x} \left(\left(\log{\left(x^{x} \right)} + 1\right) \left(x^{x}\right)^{x^{x}} \log{\left(x \right)} + \frac{x^{x^{x}}}{x}\right)\right) \log{\left(x \right)} + \frac{x^{x} x^{x^{x}} \log{\left(x \right)}}{x}

  2. Simplificamos:

    (xx(x(log⁡(xx)+1)(xx)xxlog⁡(x)+xxx)+xx+xx+xx+xx+1(log⁡(x)+1)log⁡(x))log⁡(x)x\frac{\left(x^{x} \left(x \left(\log{\left(x^{x} \right)} + 1\right) \left(x^{x}\right)^{x^{x}} \log{\left(x \right)} + x^{x^{x}}\right) + x^{x + x^{x}} + x^{x + x^{x} + 1} \left(\log{\left(x \right)} + 1\right) \log{\left(x \right)}\right) \log{\left(x \right)}}{x}

Respuesta:

(xx(x(log⁡(xx)+1)(xx)xxlog⁡(x)+xxx)+xx+xx+xx+xx+1(log⁡(x)+1)log⁡(x))log⁡(x)x\frac{\left(x^{x} \left(x \left(\log{\left(x^{x} \right)} + 1\right) \left(x^{x}\right)^{x^{x}} \log{\left(x \right)} + x^{x^{x}}\right) + x^{x + x^{x}} + x^{x + x^{x} + 1} \left(\log{\left(x \right)} + 1\right) \log{\left(x \right)}\right) \log{\left(x \right)}}{x}

Primera derivada [src] 
/   / / x\                                             \                               \              / x\       
|   | \x /    / x\ / x                         \       |       / x\                    |           x  \x /       
| x |x        \x / |x     x                    |       |    x  \x /                    |          x *x    *log(x)
|x *|----- + x    *|-- + x *(1 + log(x))*log(x)|*log(x)| + x *x    *(1 + log(x))*log(x)|*log(x) + ---------------
\   \  x           \x                          /       /                               /                 x
(xxxxx(log⁡(x)+1)log⁡(x)+xx(xxx(xx(log⁡(x)+1)log⁡(x)+xxx)log⁡(x)+xxxx))log⁡(x)+xxxxxlog⁡(x)x\left(x^{x} x^{x^{x}} \left(\log{\left(x \right)} + 1\right) \log{\left(x \right)} + x^{x} \left(x^{x^{x}} \left(x^{x} \left(\log{\left(x \right)} + 1\right) \log{\left(x \right)} + \frac{x^{x}}{x}\right) \log{\left(x \right)} + \frac{x^{x^{x}}}{x}\right)\right) \log{\left(x \right)} + \frac{x^{x} x^{x^{x}} \log{\left(x \right)}}{x}
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Segunda derivada [src] 
         //                                                                                                                                                                                                      x /1                      \\                     /1                          x /1                      \       \\
    / x\ ||                                                                                                  /                                            2                                        \          2*x *|- + (1 + log(x))*log(x)||                   2*|- + (1 + log(x))*log(x) + x *|- + (1 + log(x))*log(x)|*log(x)||
 x  \x / ||  1    /1               2\                         /1    x /1                      \       \    x |  1    log(x)    x /1                      \                2          2*(1 + log(x))|               \x                      /|          log(x)     \x                            \x                      /       /|
x *x    *||- -- + |- + (1 + log(x)) |*log(x) + 2*(1 + log(x))*|- + x *|- + (1 + log(x))*log(x)|*log(x)| + x *|- -- + ------ + x *|- + (1 + log(x))*log(x)|  + (1 + log(x)) *log(x) + --------------|*log(x) + ------------------------------|*log(x) - ------ + -----------------------------------------------------------------|
         ||   2   \x                /                         \x      \x                      /       /      |   2     x         \x                      /                                 x       |                        x               |             2                                     x                                |
         \\  x                                                                                               \  x                                                                                  /                                        /            x                                                                       /
xxxxx((xx(xx((log⁡(x)+1)log⁡(x)+1x)2+(log⁡(x)+1)2log⁡(x)+2(log⁡(x)+1)x+log⁡(x)x−1x2)log⁡(x)+2(xx((log⁡(x)+1)log⁡(x)+1x)log⁡(x)+1x)(log⁡(x)+1)+((log⁡(x)+1)2+1x)log⁡(x)+2xx((log⁡(x)+1)log⁡(x)+1x)x−1x2)log⁡(x)+2(xx((log⁡(x)+1)log⁡(x)+1x)log⁡(x)+(log⁡(x)+1)log⁡(x)+1x)x−log⁡(x)x2)x^{x} x^{x^{x}} \left(\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) \log{\left(x \right)} + 2 \left(x^{x} \left(\left(\log{\left(x \right)} + 1\right) \log{\left(x \right)} + \frac{1}{x}\right) \log{\left(x \right)} + \frac{1}{x}\right) \left(\log{\left(x \right)} + 1\right) + \left(\left(\log{\left(x \right)} + 1\right)^{2} + \frac{1}{x}\right) \log{\left(x \right)} + \frac{2 x^{x} \left(\left(\log{\left(x \right)} + 1\right) \log{\left(x \right)} + \frac{1}{x}\right)}{x} - \frac{1}{x^{2}}\right) \log{\left(x \right)} + \frac{2 \left(x^{x} \left(\left(\log{\left(x \right)} + 1\right) \log{\left(x \right)} + \frac{1}{x}\right) \log{\left(x \right)} + \left(\log{\left(x \right)} + 1\right) \log{\left(x \right)} + \frac{1}{x}\right)}{x} - \frac{\log{\left(x \right)}}{x^{2}}\right)
Simplificar
Tercera derivada [src] 
         /                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         /                                                                                                                                                                                                      x /1                      \\\
         |/                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     /                                            2                                        \\                                                                                           |                                                                                                  /                                            2                                        \          2*x *|- + (1 + log(x))*log(x)|||
         ||                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   x |  1    log(x)    x /1                      \                2          2*(1 + log(x))||                                                                                           |  1    /1               2\                         /1    x /1                      \       \    x |  1    log(x)    x /1                      \                2          2*(1 + log(x))|               \x                      /||
         ||                                                                   /                                                                                                              x /1                      \\                                                                                                                                                                                                                                                                                                                    x /1                      \   3*x *|- -- + ------ + x *|- + (1 + log(x))*log(x)|  + (1 + log(x)) *log(x) + --------------||            /1                          x /1                      \       \              3*|- -- + |- + (1 + log(x)) |*log(x) + 2*(1 + log(x))*|- + x *|- + (1 + log(x))*log(x)|*log(x)| + x *|- -- + ------ + x *|- + (1 + log(x))*log(x)|  + (1 + log(x)) *log(x) + --------------|*log(x) + ------------------------------||
    / x\ ||                                                                   |          /                                            2                                        \          2*x *|- + (1 + log(x))*log(x)||                                                                        /                                        3                                                                  2                                                                                                                 \          3*x *|- + (1 + log(x))*log(x)|        |   2     x         \x                      /                                 x       ||          3*|- + (1 + log(x))*log(x) + x *|- + (1 + log(x))*log(x)|*log(x)|                |   2   \x                /                         \x      \x                      /       /      |   2     x         \x                      /                                 x       |                        x               ||
 x  \x / ||2    /            3   1    3*(1 + log(x))\                         |  1     x |  1    log(x)    x /1                      \                2          2*(1 + log(x))|               \x                      /|     /1               2\ /1    x /1                      \       \    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                      /       /   2*log(x)     \  x                                                                                               \  x                                                                                  /                                        /|
x *x    *||-- + |(1 + log(x))  - -- + --------------|*log(x) + 3*(1 + log(x))*|- -- + x *|- -- + ------ + x *|- + (1 + log(x))*log(x)|  + (1 + log(x)) *log(x) + --------------|*log(x) + ------------------------------| + 3*|- + (1 + log(x)) |*|- + x *|- + (1 + log(x))*log(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) + --------------||*log(x) - ------------------------------ + --------------------------------------------------------------------------------------------|*log(x) - ----------------------------------------------------------------- + -------- + -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|
         || 3   |                 2         x       |                         |   2      |   2     x         \x                      /                                 x       |                        x               |     \x                / \x      \x                      /       /      | 3    2        \x                      /                              2            2                x                    x                  \x                      / |   2     x                                   x       ||                         2                                                              x                                              |                                           2                                      3                                                                                                                        x                                                                                                                  |
         \\x    \                x                  /                         \  x       \  x                                                                                  /                                        /                                                                        \x    x                                                               x            x                                                                                   \  x                                                  //                        x                                                                                                              /                                          x                                      x                                                                                                                                                                                                                                            /
xxxxx((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)x−1x2)+(log⁡(x)+1)3log⁡(x)+3(log⁡(x)+1)2x+3(log⁡(x)+1)log⁡(x)x−3(log⁡(x)+1)x2−log⁡(x)x2+3x2+2x3)log⁡(x)+3(xx((log⁡(x)+1)log⁡(x)+1x)log⁡(x)+1x)((log⁡(x)+1)2+1x)+3(log⁡(x)+1)(xx(xx((log⁡(x)+1)log⁡(x)+1x)2+(log⁡(x)+1)2log⁡(x)+2(log⁡(x)+1)x+log⁡(x)x−1x2)log⁡(x)+2xx((log⁡(x)+1)log⁡(x)+1x)x−1x2)+((log⁡(x)+1)3+3(log⁡(x)+1)x−1x2)log⁡(x)+3xx(xx((log⁡(x)+1)log⁡(x)+1x)2+(log⁡(x)+1)2log⁡(x)+2(log⁡(x)+1)x+log⁡(x)x−1x2)x−3xx((log⁡(x)+1)log⁡(x)+1x)x2+2x3)log⁡(x)+3(xx(xx((log⁡(x)+1)log⁡(x)+1x)2+(log⁡(x)+1)2log⁡(x)+2(log⁡(x)+1)x+log⁡(x)x−1x2)log⁡(x)+2(xx((log⁡(x)+1)log⁡(x)+1x)log⁡(x)+1x)(log⁡(x)+1)+((log⁡(x)+1)2+1x)log⁡(x)+2xx((log⁡(x)+1)log⁡(x)+1x)x−1x2)x−3(xx((log⁡(x)+1)log⁡(x)+1x)log⁡(x)+(log⁡(x)+1)log⁡(x)+1x)x2+2log⁡(x)x3)x^{x} x^{x^{x}} \left(\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) \log{\left(x \right)} + 3 \left(x^{x} \left(\left(\log{\left(x \right)} + 1\right) \log{\left(x \right)} + \frac{1}{x}\right) \log{\left(x \right)} + \frac{1}{x}\right) \left(\left(\log{\left(x \right)} + 1\right)^{2} + \frac{1}{x}\right) + 3 \left(\log{\left(x \right)} + 1\right) \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) \log{\left(x \right)} + \frac{2 x^{x} \left(\left(\log{\left(x \right)} + 1\right) \log{\left(x \right)} + \frac{1}{x}\right)}{x} - \frac{1}{x^{2}}\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) \log{\left(x \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{3 x^{x} \left(\left(\log{\left(x \right)} + 1\right) \log{\left(x \right)} + \frac{1}{x}\right)}{x^{2}} + \frac{2}{x^{3}}\right) \log{\left(x \right)} + \frac{3 \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) \log{\left(x \right)} + 2 \left(x^{x} \left(\left(\log{\left(x \right)} + 1\right) \log{\left(x \right)} + \frac{1}{x}\right) \log{\left(x \right)} + \frac{1}{x}\right) \left(\log{\left(x \right)} + 1\right) + \left(\left(\log{\left(x \right)} + 1\right)^{2} + \frac{1}{x}\right) \log{\left(x \right)} + \frac{2 x^{x} \left(\left(\log{\left(x \right)} + 1\right) \log{\left(x \right)} + \frac{1}{x}\right)}{x} - \frac{1}{x^{2}}\right)}{x} - \frac{3 \left(x^{x} \left(\left(\log{\left(x \right)} + 1\right) \log{\left(x \right)} + \frac{1}{x}\right) \log{\left(x \right)} + \left(\log{\left(x \right)} + 1\right) \log{\left(x \right)} + \frac{1}{x}\right)}{x^{2}} + \frac{2 \log{\left(x \right)}}{x^{3}}\right)
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