Производная (cot(x))^(atan(x)+1)

Вы ввели

   atan(x) + 1   
cot           (x)

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

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

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

2. Теперь упростим:

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

---

Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

                  /                                               3                                                                   /                                               2                                                  \                                                                           2                  3                                2                                                     \
                  |  /                /       2   \              \                      /                /       2   \              \ |                                  /       2   \                                      /       2   \|     /       2   \                                            /       2   \      /       2   \                    /       2   \                     2                   /       2   \|
   1 + atan(x)    |  |  log(cot(x))   \1 + cot (x)/*(1 + atan(x))|    2*log(cot(x))     |  log(cot(x))   \1 + cot (x)/*(1 + atan(x))| |    /       2   \                 \1 + cot (x)/ *(1 + atan(x))   2*x*log(cot(x))   2*\1 + cot (x)/|   6*\1 + cot (x)/     /       2   \                        3*\1 + cot (x)/    2*\1 + cot (x)/ *(1 + atan(x))   4*\1 + cot (x)/ *(1 + atan(x))   8*x *log(cot(x))   6*x*\1 + cot (x)/|
cot           (x)*|- |- ----------- + ---------------------------|  - ------------- + 3*|- ----------- + ---------------------------|*|- 2*\1 + cot (x)/*(1 + atan(x)) + ---------------------------- + --------------- + ---------------| + --------------- - 4*\1 + cot (x)/*(1 + atan(x))*cot(x) - ---------------- - ------------------------------ + ------------------------------ + ---------------- + -----------------|
                  |  |          2                cot(x)          |              2       |          2                cot(x)          | |                                               2                            2      /     2\       |             2                                              /     2\    2                    3                              cot(x)                          3                2       |
                  |  \     1 + x                                 /      /     2\        \     1 + x                                 / |                                            cot (x)                 /     2\       \1 + x /*cot(x)|        1 + x                                               \1 + x /*cot (x)              cot (x)                                                   /     2\         /     2\        |
                  \                                                     \1 + x /                                                      \                                                                    \1 + x /                      /                                                                                                                                                    \1 + x /         \1 + x / *cot(x)/

$$\left(\frac{8 x^{2}}{\left(x^{2} + 1\right)^{3}} \log{\left (\cot{\left (x \right )} \right )} + \frac{6 x \left(\cot^{2}{\left (x \right )} + 1\right)}{\left(x^{2} + 1\right)^{2} \cot{\left (x \right )}} - \left(\frac{1}{\cot{\left (x \right )}} \left(\cot^{2}{\left (x \right )} + 1\right) \left(\operatorname{atan}{\left (x \right )} + 1\right) - \frac{1}{x^{2} + 1} \log{\left (\cot{\left (x \right )} \right )}\right)^{3} + 3 \left(\frac{1}{\cot{\left (x \right )}} \left(\cot^{2}{\left (x \right )} + 1\right) \left(\operatorname{atan}{\left (x \right )} + 1\right) - \frac{1}{x^{2} + 1} \log{\left (\cot{\left (x \right )} \right )}\right) \left(\frac{2 x}{\left(x^{2} + 1\right)^{2}} \log{\left (\cot{\left (x \right )} \right )} + \frac{\left(\cot^{2}{\left (x \right )} + 1\right)^{2}}{\cot^{2}{\left (x \right )}} \left(\operatorname{atan}{\left (x \right )} + 1\right) - 2 \left(\cot^{2}{\left (x \right )} + 1\right) \left(\operatorname{atan}{\left (x \right )} + 1\right) + \frac{2 \cot^{2}{\left (x \right )} + 2}{\left(x^{2} + 1\right) \cot{\left (x \right )}}\right) - \frac{2 \left(\cot^{2}{\left (x \right )} + 1\right)^{3}}{\cot^{3}{\left (x \right )}} \left(\operatorname{atan}{\left (x \right )} + 1\right) + \frac{4 \left(\cot^{2}{\left (x \right )} + 1\right)^{2}}{\cot{\left (x \right )}} \left(\operatorname{atan}{\left (x \right )} + 1\right) - 4 \left(\cot^{2}{\left (x \right )} + 1\right) \left(\operatorname{atan}{\left (x \right )} + 1\right) \cot{\left (x \right )} - \frac{3 \left(\cot^{2}{\left (x \right )} + 1\right)^{2}}{\left(x^{2} + 1\right) \cot^{2}{\left (x \right )}} + \frac{1}{x^{2} + 1} \left(6 \cot^{2}{\left (x \right )} + 6\right) - \frac{2}{\left(x^{2} + 1\right)^{2}} \log{\left (\cot{\left (x \right )} \right )}\right) \cot^{\operatorname{atan}{\left (x \right )} + 1}{\left (x \right )}$$