Производная (log(x)+1)^acot(x)

Вы ввели

            acot(x)
(log(x) + 1)       

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

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

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

Вторая производная

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

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

Третья производная

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

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