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

Solución

Ha introducido [src]

  x ________
x*\/ log(x)

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

x*log(x)^(1/x)

Solución detallada

Se aplica la regla de la derivada de una multiplicación:

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

$f{\left(x \right)} = x$ ; calculamos $\frac{d}{d x} f{\left(x \right)}$ :
1. Según el principio, aplicamos: $x$ tenemos $1$
$g{\left(x \right)} = \log{\left(x \right)}^{\frac{1}{x}}$ ; calculamos $\frac{d}{d x} g{\left(x \right)}$ :
1. No logro encontrar los pasos en la búsqueda de esta derivada.
  
  Perola derivada
  
  $\left(\log{\left(\frac{1}{x} \right)} + 1\right) \left(\frac{1}{x}\right)^{\frac{1}{x}}$
Como resultado de: $x \left(\log{\left(\frac{1}{x} \right)} + 1\right) \left(\frac{1}{x}\right)^{\frac{1}{x}} + \log{\left(x \right)}^{\frac{1}{x}}$

Respuesta:

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

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

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

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

Simplificar

Segunda derivada [src]

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

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

Simplificar

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

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

Simplificar

Gráfico

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

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