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Derivada de la función/ y=(lnx)/ y=(lnx)^(ex)

Derivada de y=(lnx)^(ex)

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

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Solución

Ha introducido [src] 
        / x\
        \E /
(log(x))
$$\log{\left(x \right)}^{e^{x}}$$ 
log(x)^(E^x)
Solución detallada

  1. No logro encontrar los pasos en la búsqueda de esta derivada.

    Perola derivada

  2. Simplificamos:

Respuesta:

Primera derivada [src] 
        / x\ /                     x   \
        \E / | x                  e    |
(log(x))    *|e *log(log(x)) + --------|
             \                 x*log(x)/
$$\left(e^{x} \log{\left(\log{\left(x \right)} \right)} + \frac{e^{x}}{x \log{\left(x \right)}}\right) \log{\left(x \right)}^{e^{x}}$$
Simplificar
Segunda derivada [src] 
        / x\ /                        2                                                     \   
        \e / |/   1                  \   x       1           1           2                  |  x
(log(x))    *||-------- + log(log(x))| *e  - --------- - ---------- + -------- + log(log(x))|*e 
             |\x*log(x)              /        2           2    2      x*log(x)              |   
             \                               x *log(x)   x *log (x)                         /
$$\left(\left(\log{\left(\log{\left(x \right)} \right)} + \frac{1}{x \log{\left(x \right)}}\right)^{2} e^{x} + \log{\left(\log{\left(x \right)} \right)} + \frac{2}{x \log{\left(x \right)}} - \frac{1}{x^{2} \log{\left(x \right)}} - \frac{1}{x^{2} \log{\left(x \right)}^{2}}\right) e^{x} \log{\left(x \right)}^{e^{x}}$$
Simplificar
Tercera derivada [src] 
        / x\ /                        3                                                                                                                                                                                 \   
        \e / |/   1                  \   2*x       3           3            2           2           3           3          /   1                  \ /      1           1           2                  \  x              |  x
(log(x))    *||-------- + log(log(x))| *e    - --------- - ---------- + --------- + ---------- + -------- + ---------- + 3*|-------- + log(log(x))|*|- --------- - ---------- + -------- + log(log(x))|*e  + log(log(x))|*e 
             |\x*log(x)              /          2           2    2       3           3    3      x*log(x)    3    2        \x*log(x)              / |   2           2    2      x*log(x)              |                 |   
             \                                 x *log(x)   x *log (x)   x *log(x)   x *log (x)              x *log (x)                              \  x *log(x)   x *log (x)                         /                 /
$$\left(\left(\log{\left(\log{\left(x \right)} \right)} + \frac{1}{x \log{\left(x \right)}}\right)^{3} e^{2 x} + 3 \left(\log{\left(\log{\left(x \right)} \right)} + \frac{1}{x \log{\left(x \right)}}\right) \left(\log{\left(\log{\left(x \right)} \right)} + \frac{2}{x \log{\left(x \right)}} - \frac{1}{x^{2} \log{\left(x \right)}} - \frac{1}{x^{2} \log{\left(x \right)}^{2}}\right) e^{x} + \log{\left(\log{\left(x \right)} \right)} + \frac{3}{x \log{\left(x \right)}} - \frac{3}{x^{2} \log{\left(x \right)}} - \frac{3}{x^{2} \log{\left(x \right)}^{2}} + \frac{2}{x^{3} \log{\left(x \right)}} + \frac{3}{x^{3} \log{\left(x \right)}^{2}} + \frac{2}{x^{3} \log{\left(x \right)}^{3}}\right) e^{x} \log{\left(x \right)}^{e^{x}}$$
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