Производная (x+1)^acot(x)

Вы ввели

       acot(x)
(x + 1)       

$$\left(x + 1\right)^{\operatorname{acot}{\left (x \right )}}$$

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

   $\left(\log{\left (\operatorname{acot}{\left (x \right )} \right )} + 1\right) \operatorname{acot}^{\operatorname{acot}{\left (x \right )}}{\left (x \right )}$

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Ответ:

$\left(\log{\left (\operatorname{acot}{\left (x \right )} \right )} + 1\right) \operatorname{acot}^{\operatorname{acot}{\left (x \right )}}{\left (x \right )}$

Первая производная

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

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

Вторая производная

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

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

Третья производная

               /                        3                                                                                                                               2                               \
       acot(x) |  /log(1 + x)   acot(x)\    2*acot(x)   2*log(1 + x)           3             /log(1 + x)   acot(x)\ /acot(x)           2           2*x*log(1 + x)\   8*x *log(1 + x)          6*x       |
(1 + x)       *|- |---------- - -------|  + --------- + ------------ + ----------------- + 3*|---------- - -------|*|-------- + ---------------- - --------------| - --------------- + -----------------|
               |  |       2      1 + x |            3            2            2 /     2\     |       2      1 + x | |       2           /     2\             2   |              3                      2|
               |  \  1 + x             /     (1 + x)     /     2\      (1 + x) *\1 + x /     \  1 + x             / |(1 + x)    (1 + x)*\1 + x /     /     2\    |      /     2\               /     2\ |
               \                                         \1 + x /                                                   \                                \1 + x /    /      \1 + x /       (1 + x)*\1 + x / /

$$\left(x + 1\right)^{\operatorname{acot}{\left (x \right )}} \left(- \frac{8 x^{2}}{\left(x^{2} + 1\right)^{3}} \log{\left (x + 1 \right )} + \frac{6 x}{\left(x + 1\right) \left(x^{2} + 1\right)^{2}} - \left(\frac{\log{\left (x + 1 \right )}}{x^{2} + 1} - \frac{\operatorname{acot}{\left (x \right )}}{x + 1}\right)^{3} + 3 \left(\frac{\log{\left (x + 1 \right )}}{x^{2} + 1} - \frac{\operatorname{acot}{\left (x \right )}}{x + 1}\right) \left(- \frac{2 x}{\left(x^{2} + 1\right)^{2}} \log{\left (x + 1 \right )} + \frac{2}{\left(x + 1\right) \left(x^{2} + 1\right)} + \frac{\operatorname{acot}{\left (x \right )}}{\left(x + 1\right)^{2}}\right) + \frac{2}{\left(x^{2} + 1\right)^{2}} \log{\left (x + 1 \right )} + \frac{3}{\left(x + 1\right)^{2} \left(x^{2} + 1\right)} + \frac{2 \operatorname{acot}{\left (x \right )}}{\left(x + 1\right)^{3}}\right)$$