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

Solución

Ha introducido [src]

 log(x)
x      
-------
   x   
log (x)

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

x^log(x)/log(x)^x

Solución detallada

Se aplica la regla de la derivada parcial:

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

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

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

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

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

Respuesta:

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

Gráfica

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Trazar el gráfico f'(x)

Primera derivada [src]

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

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

Simplificar

Segunda derivada [src]

                 /                                                             1        /  1                 \       \
                 |                      2     /                  2   \   1 - ------   4*|------ + log(log(x))|*log(x)|
 log(x)    -x    |/  1                 \    2*\1 - log(x) + 2*log (x)/       log(x)     \log(x)              /       |
x      *log  (x)*||------ + log(log(x))|  + -------------------------- - ---------- - -------------------------------|
                 |\log(x)              /                 2                x*log(x)                   x               |
                 \                                      x                                                            /

$$x^{\log{\left(x \right)}} \left(\left(\log{\left(\log{\left(x \right)} \right)} + \frac{1}{\log{\left(x \right)}}\right)^{2} - \frac{1 - \frac{1}{\log{\left(x \right)}}}{x \log{\left(x \right)}} - \frac{4 \left(\log{\left(\log{\left(x \right)} \right)} + \frac{1}{\log{\left(x \right)}}\right) \log{\left(x \right)}}{x} + \frac{2 \left(2 \log{\left(x \right)}^{2} - \log{\left(x \right)} + 1\right)}{x^{2}}\right) \log{\left(x \right)}^{- x}$$

Simplificar

Tercera derivada [src]

                 /                                                                                                                                            /                                1   \                                               \
                 |                                                                               2                                                            |                      2   1 - ------|                                               |
                 |                                                                        1 - -------     /  1                 \ /                  2   \     |/  1                 \        log(x)|            /      1   \ /  1                 \|
                 |                        3     /          2           3              \          2      6*|------ + log(log(x))|*\1 - log(x) + 2*log (x)/   6*||------ + log(log(x))|  - ----------|*log(x)   3*|1 - ------|*|------ + log(log(x))||
 log(x)    -x    |  /  1                 \    2*\-3 - 6*log (x) + 4*log (x) + 8*log(x)/       log (x)     \log(x)              /                              \\log(x)              /     x*log(x) /            \    log(x)/ \log(x)              /|
x      *log  (x)*|- |------ + log(log(x))|  + ----------------------------------------- + ----------- - ------------------------------------------------- + ----------------------------------------------- + -------------------------------------|
                 |  \log(x)              /                         3                        2                                    2                                                 x                                         x*log(x)              |
                 \                                                x                        x *log(x)                            x                                                                                                                  /

$$x^{\log{\left(x \right)}} \left(- \left(\log{\left(\log{\left(x \right)} \right)} + \frac{1}{\log{\left(x \right)}}\right)^{3} + \frac{3 \left(1 - \frac{1}{\log{\left(x \right)}}\right) \left(\log{\left(\log{\left(x \right)} \right)} + \frac{1}{\log{\left(x \right)}}\right)}{x \log{\left(x \right)}} + \frac{6 \left(\left(\log{\left(\log{\left(x \right)} \right)} + \frac{1}{\log{\left(x \right)}}\right)^{2} - \frac{1 - \frac{1}{\log{\left(x \right)}}}{x \log{\left(x \right)}}\right) \log{\left(x \right)}}{x} + \frac{1 - \frac{2}{\log{\left(x \right)}^{2}}}{x^{2} \log{\left(x \right)}} - \frac{6 \left(\log{\left(\log{\left(x \right)} \right)} + \frac{1}{\log{\left(x \right)}}\right) \left(2 \log{\left(x \right)}^{2} - \log{\left(x \right)} + 1\right)}{x^{2}} + \frac{2 \left(4 \log{\left(x \right)}^{3} - 6 \log{\left(x \right)}^{2} + 8 \log{\left(x \right)} - 3\right)}{x^{3}}\right) \log{\left(x \right)}^{- x}$$

Simplificar

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

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