Производная x^(e^(cot(x)))

Вы ввели

 / cot(x)\
 \E      /
x         

$$x^{e^{\cot{\left (x \right )}}}$$

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

   $\left(\log{\left (e^{\cot{\left (x \right )}} \right )} + 1\right) \left(e^{\cot{\left (x \right )}}\right)^{e^{\cot{\left (x \right )}}}$

2. Теперь упростим:

   $\left(\log{\left (e^{\cot{\left (x \right )}} \right )} + 1\right) \left(e^{\cot{\left (x \right )}}\right)^{e^{\cot{\left (x \right )}}}$

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Ответ:

$\left(\log{\left (e^{\cot{\left (x \right )}} \right )} + 1\right) \left(e^{\cot{\left (x \right )}}\right)^{e^{\cot{\left (x \right )}}}$

Первая производная

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

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

Вторая производная

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

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

Третья производная

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

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