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x^(ln(exp(x)-1))
Derivada de la función/ exp(x)/ x^(ln(exp(x)-1))

Derivada de x^(ln(exp(x)-1))

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

Ha introducido [src] 
    / x    \
 log\e  - 1/
x
xlog⁡(ex−1)x^{\log{\left(e^{x} - 1 \right)}} 
x^log(exp(x) - 1)
Solución detallada

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

    Perola derivada

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

  2. Simplificamos:

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

Respuesta:

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

Gráfica
02468-8-6-4-2-1010-5000000000050000000000
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Trazar el gráfico f'(x)
Primera derivada [src] 
    / x    \ /   / x    \    x       \
 log\e  - 1/ |log\e  - 1/   e *log(x)|
x           *|----------- + ---------|
             |     x           x     |
             \                e  - 1 /
xlog⁡(ex−1)(exlog⁡(x)ex−1+log⁡(ex−1)x)x^{\log{\left(e^{x} - 1 \right)}} \left(\frac{e^{x} \log{\left(x \right)}}{e^{x} - 1} + \frac{\log{\left(e^{x} - 1 \right)}}{x}\right)
Simplificar
Segunda derivada [src] 
              /                          2                                                       \
    /      x\ |/   /      x\    x       \       /      x\    x           2*x                 x   |
 log\-1 + e / ||log\-1 + e /   e *log(x)|    log\-1 + e /   e *log(x)   e   *log(x)       2*e    |
x            *||------------ + ---------|  - ------------ + --------- - ----------- + -----------|
              ||     x                x |          2               x              2     /      x\|
              |\                -1 + e  /         x          -1 + e      /      x\    x*\-1 + e /|
              \                                                          \-1 + e /               /
xlog⁡(ex−1)((exlog⁡(x)ex−1+log⁡(ex−1)x)2+exlog⁡(x)ex−1−e2xlog⁡(x)(ex−1)2+2exx(ex−1)−log⁡(ex−1)x2)x^{\log{\left(e^{x} - 1 \right)}} \left(\left(\frac{e^{x} \log{\left(x \right)}}{e^{x} - 1} + \frac{\log{\left(e^{x} - 1 \right)}}{x}\right)^{2} + \frac{e^{x} \log{\left(x \right)}}{e^{x} - 1} - \frac{e^{2 x} \log{\left(x \right)}}{\left(e^{x} - 1\right)^{2}} + \frac{2 e^{x}}{x \left(e^{x} - 1\right)} - \frac{\log{\left(e^{x} - 1 \right)}}{x^{2}}\right)
Simplificar
Tercera derivada [src] 
              /                          3                                                                                                                                                                                                 \
    /      x\ |/   /      x\    x       \         /      x\     /   /      x\    x       \ /     /      x\    x           2*x                 x   \    x                2*x             x          2*x             3*x                 x   |
 log\-1 + e / ||log\-1 + e /   e *log(x)|    2*log\-1 + e /     |log\-1 + e /   e *log(x)| |  log\-1 + e /   e *log(x)   e   *log(x)       2*e    |   e *log(x)      3*e             3*e        3*e   *log(x)   2*e   *log(x)       3*e    |
x            *||------------ + ---------|  + -------------- + 3*|------------ + ---------|*|- ------------ + --------- - ----------- + -----------| + --------- - ------------ - ------------ - ------------- + ------------- + -----------|
              ||     x                x |           3           |     x                x | |        2               x              2     /      x\|          x               2    2 /      x\              2               3      /      x\|
              |\                -1 + e  /          x            \                -1 + e  / |       x          -1 + e      /      x\    x*\-1 + e /|    -1 + e       /      x\    x *\-1 + e /     /      x\       /      x\     x*\-1 + e /|
              \                                                                            \                              \-1 + e /               /               x*\-1 + e /                     \-1 + e /       \-1 + e /                /
xlog⁡(ex−1)((exlog⁡(x)ex−1+log⁡(ex−1)x)3+3(exlog⁡(x)ex−1+log⁡(ex−1)x)(exlog⁡(x)ex−1−e2xlog⁡(x)(ex−1)2+2exx(ex−1)−log⁡(ex−1)x2)+exlog⁡(x)ex−1−3e2xlog⁡(x)(ex−1)2+2e3xlog⁡(x)(ex−1)3+3exx(ex−1)−3e2xx(ex−1)2−3exx2(ex−1)+2log⁡(ex−1)x3)x^{\log{\left(e^{x} - 1 \right)}} \left(\left(\frac{e^{x} \log{\left(x \right)}}{e^{x} - 1} + \frac{\log{\left(e^{x} - 1 \right)}}{x}\right)^{3} + 3 \left(\frac{e^{x} \log{\left(x \right)}}{e^{x} - 1} + \frac{\log{\left(e^{x} - 1 \right)}}{x}\right) \left(\frac{e^{x} \log{\left(x \right)}}{e^{x} - 1} - \frac{e^{2 x} \log{\left(x \right)}}{\left(e^{x} - 1\right)^{2}} + \frac{2 e^{x}}{x \left(e^{x} - 1\right)} - \frac{\log{\left(e^{x} - 1 \right)}}{x^{2}}\right) + \frac{e^{x} \log{\left(x \right)}}{e^{x} - 1} - \frac{3 e^{2 x} \log{\left(x \right)}}{\left(e^{x} - 1\right)^{2}} + \frac{2 e^{3 x} \log{\left(x \right)}}{\left(e^{x} - 1\right)^{3}} + \frac{3 e^{x}}{x \left(e^{x} - 1\right)} - \frac{3 e^{2 x}}{x \left(e^{x} - 1\right)^{2}} - \frac{3 e^{x}}{x^{2} \left(e^{x} - 1\right)} + \frac{2 \log{\left(e^{x} - 1 \right)}}{x^{3}}\right)
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Gráfico
Derivada de x^(ln(exp(x)-1))
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