Производная x^e^acot(x)

Вы ввели

 / acot(x)\
 \e       /
x          

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

  / / acot(x)\\
d | \e       /|
--\x          /
dx             

$$\frac{d}{d x} x^{e^{\operatorname{acot}{\left(x \right)}}}$$

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

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

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Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

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

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

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