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x^(log(x,exp))
Derivada de la función/ x^(log(x,exp))

Derivada de x^(log(x,exp))

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

Ha introducido [src] 
  log(x)
 -------
    / x\
 log\e /
x
xlog⁡(x)log⁡(ex)x^{\frac{\log{\left(x \right)}}{\log{\left(e^{x} \right)}}} 
x^(log(x)/log(exp(x)))
Solución detallada

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

    Perola derivada

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

  2. Simplificamos:

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

Respuesta:

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

Gráfica
02468-8-6-4-2-1010-5e255e25
Trazar el gráfico f(x)
Trazar el gráfico f'(x)
Primera derivada [src] 
  log(x)                                            
 -------                                            
    / x\                                            
 log\e / //    1        log(x) \            log(x) \
x       *||--------- - --------|*log(x) + ---------|
         ||     / x\      2/ x\|               / x\|
         \\x*log\e /   log \e //          x*log\e //
xlog⁡(x)log⁡(ex)((−log⁡(x)log⁡(ex)2+1xlog⁡(ex))log⁡(x)+log⁡(x)xlog⁡(ex))x^{\frac{\log{\left(x \right)}}{\log{\left(e^{x} \right)}}} \left(\left(- \frac{\log{\left(x \right)}}{\log{\left(e^{x} \right)}^{2}} + \frac{1}{x \log{\left(e^{x} \right)}}\right) \log{\left(x \right)} + \frac{\log{\left(x \right)}}{x \log{\left(e^{x} \right)}}\right)
Simplificar
Segunda derivada [src] 
         /                                                                                  2                    \
  log(x) |       1    log(x)                                                 /  2    log(x)\     2               |
 ------- |     - - + -------                                                 |- - + -------| *log (x)            |
    / x\ |       x      / x\                                                 |  x      / x\|                     |
 log\e / |1          log\e /   log(x)   /1    2*log(x)       2    \          \      log\e //              log(x) |
x       *|-- - ------------- - ------ - |-- - -------- + ---------|*log(x) + ------------------------ - ---------|
         | 2         x            2     | 2      2/ x\        / x\|                     / x\                 / x\|
         \x                      x      \x    log \e /   x*log\e //                  log\e /            x*log\e //
------------------------------------------------------------------------------------------------------------------
                                                        / x\                                                      
                                                     log\e /
xlog⁡(x)log⁡(ex)((log⁡(x)log⁡(ex)−2x)2log⁡(x)2log⁡(ex)−(−2log⁡(x)log⁡(ex)2+2xlog⁡(ex)+1x2)log⁡(x)−log⁡(x)log⁡(ex)−1xx−log⁡(x)xlog⁡(ex)−log⁡(x)x2+1x2)log⁡(ex)\frac{x^{\frac{\log{\left(x \right)}}{\log{\left(e^{x} \right)}}} \left(\frac{\left(\frac{\log{\left(x \right)}}{\log{\left(e^{x} \right)}} - \frac{2}{x}\right)^{2} \log{\left(x \right)}^{2}}{\log{\left(e^{x} \right)}} - \left(- \frac{2 \log{\left(x \right)}}{\log{\left(e^{x} \right)}^{2}} + \frac{2}{x \log{\left(e^{x} \right)}} + \frac{1}{x^{2}}\right) \log{\left(x \right)} - \frac{\frac{\log{\left(x \right)}}{\log{\left(e^{x} \right)}} - \frac{1}{x}}{x} - \frac{\log{\left(x \right)}}{x \log{\left(e^{x} \right)}} - \frac{\log{\left(x \right)}}{x^{2}} + \frac{1}{x^{2}}\right)}{\log{\left(e^{x} \right)}}
Simplificar
Tercera derivada [src] 
         /                                                                                                                                                                                                         /         1    log(x)                                                          \       \
         |                                                                                                                                                                                                         |       - - + -------                                                          |       |
         |                                                                                                                                                 3                                                       |         x      / x\                                                          |       |
  log(x) |         1    log(x)                                                        /1    2*log(x)       2    \                           /  2    log(x)\     3                                  /  2    log(x)\ |  1          log\e /   log(x)   /1    2*log(x)       2    \            log(x) |       |
 ------- |       - - + -------                                                      2*|-- - -------- + ---------|                           |- - + -------| *log (x)                             3*|- - + -------|*|- -- + ------------- + ------ + |-- - -------- + ---------|*log(x) + ---------|*log(x)|
    / x\ |         x      / x\                                                        | 2      2/ x\        / x\|                           |  x      / x\|                                        |  x      / x\| |   2         x            2     | 2      2/ x\        / x\|               / x\|       |
 log\e / |  3          log\e /   /2    6*log(x)       3            6     \            \x    log \e /   x*log\e //       2        2*log(x)   \      log\e //             2*log(x)     2*log(x)      \      log\e // \  x                      x      \x    log \e /   x*log\e //          x*log\e //       |
x       *|- -- + ------------- + |-- - -------- + ---------- + ----------|*log(x) - ----------------------------- - ---------- + -------- - ------------------------ + ---------- + ---------- + ---------------------------------------------------------------------------------------------------------|
         |   3          2        | 3      3/ x\    2    / x\        2/ x\|                        x                  2    / x\       3                 2/ x\                2/ x\    2    / x\                                                       / x\                                                 |
         \  x          x         \x    log \e /   x *log\e /   x*log \e //                                          x *log\e /      x               log \e /           x*log \e /   x *log\e /                                                    log\e /                                                 /
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                                     / x\                                                                                                                                                  
                                                                                                                                                  log\e /
xlog⁡(x)log⁡(ex)(−(log⁡(x)log⁡(ex)−2x)3log⁡(x)3log⁡(ex)2+3(log⁡(x)log⁡(ex)−2x)((−2log⁡(x)log⁡(ex)2+2xlog⁡(ex)+1x2)log⁡(x)+log⁡(x)log⁡(ex)−1xx+log⁡(x)xlog⁡(ex)+log⁡(x)x2−1x2)log⁡(x)log⁡(ex)+(−6log⁡(x)log⁡(ex)3+6xlog⁡(ex)2+3x2log⁡(ex)+2x3)log⁡(x)−2(−2log⁡(x)log⁡(ex)2+2xlog⁡(ex)+1x2)x+2log⁡(x)xlog⁡(ex)2+log⁡(x)log⁡(ex)−1xx2+2log⁡(x)x2log⁡(ex)−2x2log⁡(ex)+2log⁡(x)x3−3x3)log⁡(ex)\frac{x^{\frac{\log{\left(x \right)}}{\log{\left(e^{x} \right)}}} \left(- \frac{\left(\frac{\log{\left(x \right)}}{\log{\left(e^{x} \right)}} - \frac{2}{x}\right)^{3} \log{\left(x \right)}^{3}}{\log{\left(e^{x} \right)}^{2}} + \frac{3 \left(\frac{\log{\left(x \right)}}{\log{\left(e^{x} \right)}} - \frac{2}{x}\right) \left(\left(- \frac{2 \log{\left(x \right)}}{\log{\left(e^{x} \right)}^{2}} + \frac{2}{x \log{\left(e^{x} \right)}} + \frac{1}{x^{2}}\right) \log{\left(x \right)} + \frac{\frac{\log{\left(x \right)}}{\log{\left(e^{x} \right)}} - \frac{1}{x}}{x} + \frac{\log{\left(x \right)}}{x \log{\left(e^{x} \right)}} + \frac{\log{\left(x \right)}}{x^{2}} - \frac{1}{x^{2}}\right) \log{\left(x \right)}}{\log{\left(e^{x} \right)}} + \left(- \frac{6 \log{\left(x \right)}}{\log{\left(e^{x} \right)}^{3}} + \frac{6}{x \log{\left(e^{x} \right)}^{2}} + \frac{3}{x^{2} \log{\left(e^{x} \right)}} + \frac{2}{x^{3}}\right) \log{\left(x \right)} - \frac{2 \left(- \frac{2 \log{\left(x \right)}}{\log{\left(e^{x} \right)}^{2}} + \frac{2}{x \log{\left(e^{x} \right)}} + \frac{1}{x^{2}}\right)}{x} + \frac{2 \log{\left(x \right)}}{x \log{\left(e^{x} \right)}^{2}} + \frac{\frac{\log{\left(x \right)}}{\log{\left(e^{x} \right)}} - \frac{1}{x}}{x^{2}} + \frac{2 \log{\left(x \right)}}{x^{2} \log{\left(e^{x} \right)}} - \frac{2}{x^{2} \log{\left(e^{x} \right)}} + \frac{2 \log{\left(x \right)}}{x^{3}} - \frac{3}{x^{3}}\right)}{\log{\left(e^{x} \right)}}
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Gráfico
Derivada de x^(log(x,exp))
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