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x(lnx)^(1/x)

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

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

Ha introducido [src]
  x ________
x*\/ log(x) 
xlog(x)1xx \log{\left(x \right)}^{\frac{1}{x}}
x*log(x)^(1/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)=xf{\left(x \right)} = x; calculamos ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. Según el principio, aplicamos: xx tenemos 11

    g(x)=log(x)1xg{\left(x \right)} = \log{\left(x \right)}^{\frac{1}{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

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

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


Respuesta:

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

Gráfica
02468-8-6-4-2-1010020
Primera derivada [src]
x ________     x ________ /    1       log(log(x))\
\/ log(x)  + x*\/ log(x) *|--------- - -----------|
                          | 2                2    |
                          \x *log(x)        x     /
x(log(log(x))x2+1x2log(x))log(x)1x+log(x)1xx \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}} + \log{\left(x \right)}^{\frac{1}{x}}
Segunda derivada [src]
           /                                             2\
           |                     /    1                 \ |
           |                     |- ------ + log(log(x))| |
x ________ |    1         1      \  log(x)              / |
\/ log(x) *|- ------ - ------- + -------------------------|
           |  log(x)      2                  x            |
           \           log (x)                            /
-----------------------------------------------------------
                              2                            
                             x                             
(1log(x)1log(x)2+(log(log(x))1log(x))2x)log(x)1xx2\frac{\left(- \frac{1}{\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^{2}}
Tercera derivada [src]
           /                                                     3                             2     /    1                 \ /   1                        3   \\
           |                             /    1                 \      /    1                 \    3*|- ------ + log(log(x))|*|------- - 2*log(log(x)) + ------||
           |                             |- ------ + log(log(x))|    3*|- ------ + log(log(x))|      \  log(x)              / |   2                      log(x)||
x ________ |  2         2         3      \  log(x)              /      \  log(x)              /                               \log (x)                         /|
\/ log(x) *|------ + ------- + ------- - ------------------------- + --------------------------- + -------------------------------------------------------------|
           |log(x)      3         2                   2                           x                                              x                              |
           \         log (x)   log (x)               x                                                                                                          /
-----------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                 3                                                                               
                                                                                x                                                                                
(2log(x)+3log(x)2+2log(x)3+3(log(log(x))1log(x))2x+3(log(log(x))1log(x))(2log(log(x))+3log(x)+1log(x)2)x(log(log(x))1log(x))3x2)log(x)1xx3\frac{\left(\frac{2}{\log{\left(x \right)}} + \frac{3}{\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)^{2}}{x} + \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^{3}}
Gráfico
Derivada de x(lnx)^(1/x)