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y=(x/(1+x))^x
Derivada de la función/ (1+x)/ y=(x/(1+x))^x

Derivada de y=(x/(1+x))^x

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

Ha introducido [src] 
       x
/  x  \ 
|-----| 
\1 + x/
(xx+1)x\left(\frac{x}{x + 1}\right)^{x} 
(x/(1 + x))^x
Solución detallada

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

    Perola derivada

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

Respuesta:

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

Gráfica
02468-8-6-4-2-10102-2
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Trazar el gráfico f'(x)
Primera derivada [src] 
       x                                          
/  x  \  /        /  1        x    \      /  x  \\
|-----| *|(1 + x)*|----- - --------| + log|-----||
\1 + x/  |        |1 + x          2|      \1 + x/|
         \        \        (1 + x) /             /
(xx+1)x((x+1)(−x(x+1)2+1x+1)+log⁡(xx+1))\left(\frac{x}{x + 1}\right)^{x} \left(\left(x + 1\right) \left(- \frac{x}{\left(x + 1\right)^{2}} + \frac{1}{x + 1}\right) + \log{\left(\frac{x}{x + 1} \right)}\right)
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Segunda derivada [src] 
         /                                                       x       /       x  \\
       x |                                 2              -1 + -----   2*|-1 + -----||
/  x  \  |  1     /        /  x  \     x  \       x            1 + x     \     1 + x/|
|-----| *|----- + |-1 - log|-----| + -----|  - -------- - ---------- + --------------|
\1 + x/  |1 + x   \        \1 + x/   1 + x/           2       x            1 + x     |
         \                                     (1 + x)                               /
(xx+1)x(−x(x+1)2+(xx+1−log⁡(xx+1)−1)2+2(xx+1−1)x+1+1x+1−xx+1−1x)\left(\frac{x}{x + 1}\right)^{x} \left(- \frac{x}{\left(x + 1\right)^{2}} + \left(\frac{x}{x + 1} - \log{\left(\frac{x}{x + 1} \right)} - 1\right)^{2} + \frac{2 \left(\frac{x}{x + 1} - 1\right)}{x + 1} + \frac{1}{x + 1} - \frac{\frac{x}{x + 1} - 1}{x}\right)
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Tercera derivada [src] 
         /                                                 x       /       x  \                               /                            x       /       x  \\                     x  \
       x |                           3              -1 + -----   6*|-1 + -----|                               |                     -1 + -----   2*|-1 + -----||              -1 + -----|
/  x  \  |  /        /  x  \     x  \       4            1 + x     \     1 + x/     /        /  x  \     x  \ |    1        x            1 + x     \     1 + x/|     4*x           1 + x|
|-----| *|- |-1 - log|-----| + -----|  - -------- + ---------- - -------------- + 3*|-1 - log|-----| + -----|*|- ----- + -------- + ---------- - --------------| + -------- + ----------|
\1 + x/  |  \        \1 + x/   1 + x/           2        2                 2        \        \1 + x/   1 + x/ |  1 + x          2       x            1 + x     |          3   x*(1 + x) |
         \                               (1 + x)        x           (1 + x)                                   \          (1 + x)                               /   (1 + x)              /
(xx+1)x(4x(x+1)3−(xx+1−log⁡(xx+1)−1)3+3(xx+1−log⁡(xx+1)−1)(x(x+1)2−2(xx+1−1)x+1−1x+1+xx+1−1x)−6(xx+1−1)(x+1)2−4(x+1)2+xx+1−1x(x+1)+xx+1−1x2)\left(\frac{x}{x + 1}\right)^{x} \left(\frac{4 x}{\left(x + 1\right)^{3}} - \left(\frac{x}{x + 1} - \log{\left(\frac{x}{x + 1} \right)} - 1\right)^{3} + 3 \left(\frac{x}{x + 1} - \log{\left(\frac{x}{x + 1} \right)} - 1\right) \left(\frac{x}{\left(x + 1\right)^{2}} - \frac{2 \left(\frac{x}{x + 1} - 1\right)}{x + 1} - \frac{1}{x + 1} + \frac{\frac{x}{x + 1} - 1}{x}\right) - \frac{6 \left(\frac{x}{x + 1} - 1\right)}{\left(x + 1\right)^{2}} - \frac{4}{\left(x + 1\right)^{2}} + \frac{\frac{x}{x + 1} - 1}{x \left(x + 1\right)} + \frac{\frac{x}{x + 1} - 1}{x^{2}}\right)
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Gráfico
Derivada de y=(x/(1+x))^x
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