Производная (x+1)^(2/x)

Вы ввели

       2
       -
       x
(x + 1) 

$$\left(x + 1\right)^{\frac{2}{x}}$$

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

   $\left(\frac{2}{x}\right)^{\frac{2}{x}} \left(\log{\left (\frac{2}{x} \right )} + 1\right)$

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Ответ:

$\left(\frac{2}{x}\right)^{\frac{2}{x}} \left(\log{\left (\frac{2}{x} \right )} + 1\right)$

Первая производная

       2                             
       -                             
       x /  2*log(x + 1)       2    \
(x + 1) *|- ------------ + ---------|
         |        2        x*(x + 1)|
         \       x                  /

$$\left(x + 1\right)^{\frac{2}{x}} \left(\frac{2}{x \left(x + 1\right)} - \frac{2}{x^{2}} \log{\left (x + 1 \right )}\right)$$

Вторая производная

           /                                               2               \
         2 |                           /  1     log(1 + x)\                |
         - |                         2*|----- - ----------|                |
         x |     1           2         \1 + x       x     /    2*log(1 + x)|
2*(1 + x) *|- -------- - --------- + ----------------------- + ------------|
           |         2   x*(1 + x)              x                    2     |
           \  (1 + x)                                               x      /
----------------------------------------------------------------------------
                                     x                                      

$$\frac{2}{x} \left(x + 1\right)^{\frac{2}{x}} \left(- \frac{1}{\left(x + 1\right)^{2}} + \frac{2}{x} \left(\frac{1}{x + 1} - \frac{1}{x} \log{\left (x + 1 \right )}\right)^{2} - \frac{2}{x \left(x + 1\right)} + \frac{2}{x^{2}} \log{\left (x + 1 \right )}\right)$$

Третья производная

           /                                                             3                  /  1     log(1 + x)\ /   1       2*log(1 + x)       2    \\
         2 |                                         /  1     log(1 + x)\                 6*|----- - ----------|*|-------- - ------------ + ---------||
         - |                                       4*|----- - ----------|                   \1 + x       x     / |       2         2        x*(1 + x)||
         x |   2       6*log(1 + x)       3          \1 + x       x     /        6                               \(1 + x)         x                  /|
2*(1 + x) *|-------- - ------------ + ---------- + ----------------------- + ---------- - ------------------------------------------------------------|
           |       3         3                 2               2              2                                        x                              |
           \(1 + x)         x         x*(1 + x)               x              x *(1 + x)                                                               /
-------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                           x                                                                           

$$\frac{2}{x} \left(x + 1\right)^{\frac{2}{x}} \left(\frac{2}{\left(x + 1\right)^{3}} - \frac{6}{x} \left(\frac{1}{x + 1} - \frac{1}{x} \log{\left (x + 1 \right )}\right) \left(\frac{1}{\left(x + 1\right)^{2}} + \frac{2}{x \left(x + 1\right)} - \frac{2}{x^{2}} \log{\left (x + 1 \right )}\right) + \frac{3}{x \left(x + 1\right)^{2}} + \frac{4}{x^{2}} \left(\frac{1}{x + 1} - \frac{1}{x} \log{\left (x + 1 \right )}\right)^{3} + \frac{6}{x^{2} \left(x + 1\right)} - \frac{6}{x^{3}} \log{\left (x + 1 \right )}\right)$$