Sr Examen

Derivada de y=e^lnx^x

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

Gráfico:

interior superior

Definida a trozos:

Solución

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

  2. Derivado es.

  3. Luego se aplica una cadena de reglas. Multiplicamos por :

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

      Perola derivada

    Como resultado de la secuencia de reglas:


Respuesta:

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