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(x^x-1)/(ln(x)^x)

Derivada de (x^x-1)/(ln(x)^x)

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

Ha introducido [src]
  x    
 x  - 1
-------
   x   
log (x)
xx1log(x)x\frac{x^{x} - 1}{\log{\left(x \right)}^{x}}
(x^x - 1)/log(x)^x
Solución detallada
  1. Se aplica la regla de la derivada parcial:

    ddxf(x)g(x)=f(x)ddxg(x)+g(x)ddxf(x)g2(x)\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(x)=xx1f{\left(x \right)} = x^{x} - 1 y g(x)=log(x)xg{\left(x \right)} = \log{\left(x \right)}^{x}.

    Para calcular ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. diferenciamos xx1x^{x} - 1 miembro por miembro:

      1. La derivada de una constante 1-1 es igual a cero.

      2. 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: xx(log(x)+1)x^{x} \left(\log{\left(x \right)} + 1\right)

    Para calcular 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)

    Ahora aplicamos la regla de la derivada de una divesión:

    (xx(xx1)(log(x)+1)+xx(log(x)+1)log(x)x)log(x)2x\left(- x^{x} \left(x^{x} - 1\right) \left(\log{\left(x \right)} + 1\right) + x^{x} \left(\log{\left(x \right)} + 1\right) \log{\left(x \right)}^{x}\right) \log{\left(x \right)}^{- 2 x}

  2. Simplificamos:

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


Respuesta:

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

Gráfica
02468-8-6-4-2-101005000000
Primera derivada [src]
 x    -x                      -x    / x    \ /    1                 \
x *log  (x)*(1 + log(x)) + log  (x)*\x  - 1/*|- ------ - log(log(x))|
                                             \  log(x)              /
xx(log(x)+1)log(x)x+(xx1)(log(log(x))1log(x))log(x)xx^{x} \left(\log{\left(x \right)} + 1\right) \log{\left(x \right)}^{- x} + \left(x^{x} - 1\right) \left(- \log{\left(\log{\left(x \right)} \right)} - \frac{1}{\log{\left(x \right)}}\right) \log{\left(x \right)}^{- x}
Segunda derivada [src]
         /                                   /                                1   \                                           \
         |                                   |                      2   1 - ------|                                           |
   -x    | x /1               2\   /      x\ |/  1                 \        log(x)|      x              /  1                 \|
log  (x)*|x *|- + (1 + log(x)) | + \-1 + x /*||------ + log(log(x))|  - ----------| - 2*x *(1 + log(x))*|------ + log(log(x))||
         \   \x                /             \\log(x)              /     x*log(x) /                     \log(x)              //
(xx((log(x)+1)2+1x)2xx(log(x)+1)(log(log(x))+1log(x))+(xx1)((log(log(x))+1log(x))211log(x)xlog(x)))log(x)x\left(x^{x} \left(\left(\log{\left(x \right)} + 1\right)^{2} + \frac{1}{x}\right) - 2 x^{x} \left(\log{\left(x \right)} + 1\right) \left(\log{\left(\log{\left(x \right)} \right)} + \frac{1}{\log{\left(x \right)}}\right) + \left(x^{x} - 1\right) \left(\left(\log{\left(\log{\left(x \right)} \right)} + \frac{1}{\log{\left(x \right)}}\right)^{2} - \frac{1 - \frac{1}{\log{\left(x \right)}}}{x \log{\left(x \right)}}\right)\right) \log{\left(x \right)}^{- x}
Tercera derivada [src]
         /                                                     /                                   2                                           \                                                                                                             \
         |                                                     |                            1 - -------     /      1   \ /  1                 \|                                                                       /                                1   \|
         |                                                     |                        3          2      3*|1 - ------|*|------ + log(log(x))||                                                                       |                      2   1 - ------||
   -x    | x /            3   1    3*(1 + log(x))\   /      x\ |  /  1                 \        log (x)     \    log(x)/ \log(x)              /|      x /1               2\ /  1                 \      x              |/  1                 \        log(x)||
log  (x)*|x *|(1 + log(x))  - -- + --------------| + \-1 + x /*|- |------ + log(log(x))|  + ----------- + -------------------------------------| - 3*x *|- + (1 + log(x)) |*|------ + log(log(x))| + 3*x *(1 + log(x))*||------ + log(log(x))|  - ----------||
         |   |                 2         x       |             |  \log(x)              /      2                          x*log(x)              |        \x                / \log(x)              /                     \\log(x)              /     x*log(x) /|
         \   \                x                  /             \                             x *log(x)                                         /                                                                                                             /
(3xx((log(x)+1)2+1x)(log(log(x))+1log(x))+3xx((log(log(x))+1log(x))211log(x)xlog(x))(log(x)+1)+xx((log(x)+1)3+3(log(x)+1)x1x2)+(xx1)((log(log(x))+1log(x))3+3(11log(x))(log(log(x))+1log(x))xlog(x)+12log(x)2x2log(x)))log(x)x\left(- 3 x^{x} \left(\left(\log{\left(x \right)} + 1\right)^{2} + \frac{1}{x}\right) \left(\log{\left(\log{\left(x \right)} \right)} + \frac{1}{\log{\left(x \right)}}\right) + 3 x^{x} \left(\left(\log{\left(\log{\left(x \right)} \right)} + \frac{1}{\log{\left(x \right)}}\right)^{2} - \frac{1 - \frac{1}{\log{\left(x \right)}}}{x \log{\left(x \right)}}\right) \left(\log{\left(x \right)} + 1\right) + x^{x} \left(\left(\log{\left(x \right)} + 1\right)^{3} + \frac{3 \left(\log{\left(x \right)} + 1\right)}{x} - \frac{1}{x^{2}}\right) + \left(x^{x} - 1\right) \left(- \left(\log{\left(\log{\left(x \right)} \right)} + \frac{1}{\log{\left(x \right)}}\right)^{3} + \frac{3 \left(1 - \frac{1}{\log{\left(x \right)}}\right) \left(\log{\left(\log{\left(x \right)} \right)} + \frac{1}{\log{\left(x \right)}}\right)}{x \log{\left(x \right)}} + \frac{1 - \frac{2}{\log{\left(x \right)}^{2}}}{x^{2} \log{\left(x \right)}}\right)\right) \log{\left(x \right)}^{- x}
Gráfico
Derivada de (x^x-1)/(ln(x)^x)