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y=(lnx)^(1/x)

Derivada de y=(lnx)^(1/x)

Función f() - derivada -er orden en el punto
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Gráfico:

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Solución

Ha introducido [src]
x ________
\/ log(x) 
$$\log{\left(x \right)}^{\frac{1}{x}}$$
log(x)^(1/x)
Solución detallada
  1. No logro encontrar los pasos en la búsqueda de esta derivada.

    Perola derivada


Respuesta:

Gráfica
Primera derivada [src]
x ________ /    1       log(log(x))\
\/ log(x) *|--------- - -----------|
           | 2                2    |
           \x *log(x)        x     /
$$\left(- \frac{\log{\left(\log{\left(x \right)} \right)}}{x^{2}} + \frac{1}{x^{2} \log{\left(x \right)}}\right) \log{\left(x \right)}^{\frac{1}{x}}$$
Segunda derivada [src]
           /                                                             2\
           |                                     /    1                 \ |
           |                                     |- ------ + log(log(x))| |
x ________ |     1        3                      \  log(x)              / |
\/ log(x) *|- ------- - ------ + 2*log(log(x)) + -------------------------|
           |     2      log(x)                               x            |
           \  log (x)                                                     /
---------------------------------------------------------------------------
                                      3                                    
                                     x                                     
$$\frac{\left(2 \log{\left(\log{\left(x \right)} \right)} - \frac{3}{\log{\left(x \right)}} - \frac{1}{\log{\left(x \right)}^{2}} + \frac{\left(\log{\left(\log{\left(x \right)} \right)} - \frac{1}{\log{\left(x \right)}}\right)^{2}}{x}\right) \log{\left(x \right)}^{\frac{1}{x}}}{x^{3}}$$
Tercera derivada [src]
           /                                                                      3     /    1                 \ /   1                        3   \\
           |                                              /    1                 \    3*|- ------ + log(log(x))|*|------- - 2*log(log(x)) + ------||
           |                                              |- ------ + log(log(x))|      \  log(x)              / |   2                      log(x)||
x ________ |                    2         6        11     \  log(x)              /                               \log (x)                         /|
\/ log(x) *|-6*log(log(x)) + ------- + ------- + ------ - ------------------------- + -------------------------------------------------------------|
           |                    3         2      log(x)                2                                            x                              |
           \                 log (x)   log (x)                        x                                                                            /
----------------------------------------------------------------------------------------------------------------------------------------------------
                                                                          4                                                                         
                                                                         x                                                                          
$$\frac{\left(- 6 \log{\left(\log{\left(x \right)} \right)} + \frac{11}{\log{\left(x \right)}} + \frac{6}{\log{\left(x \right)}^{2}} + \frac{2}{\log{\left(x \right)}^{3}} + \frac{3 \left(\log{\left(\log{\left(x \right)} \right)} - \frac{1}{\log{\left(x \right)}}\right) \left(- 2 \log{\left(\log{\left(x \right)} \right)} + \frac{3}{\log{\left(x \right)}} + \frac{1}{\log{\left(x \right)}^{2}}\right)}{x} - \frac{\left(\log{\left(\log{\left(x \right)} \right)} - \frac{1}{\log{\left(x \right)}}\right)^{3}}{x^{2}}\right) \log{\left(x \right)}^{\frac{1}{x}}}{x^{4}}$$
Gráfico
Derivada de y=(lnx)^(1/x)