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x^(ln(1+e^x))
Derivada de la función/ 1+e^x/ x^(ln(1+e^x))

Derivada de x^(ln(1+e^x))

Función f() - derivada -er orden en el punto
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Gráfico:

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Definida a trozos:

Solución

Ha introducido [src] 
    /     x\
 log\1 + E /
x
xlog⁡(ex+1)x^{\log{\left(e^{x} + 1 \right)}} 
x^log(1 + E^x)
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
02468-8-6-4-2-1010050000000000
Trazar el gráfico f(x)
Trazar el gráfico f'(x)
Primera derivada [src] 
    /     x\ /   /     x\    x       \
 log\1 + E / |log\1 + E /   e *log(x)|
x           *|----------- + ---------|
             |     x               x |
             \                1 + E  /
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)
Simplificar
Gráfico
Derivada de x^(ln(1+e^x))
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