Производная ((x))^exp(tan(x))

Вы ввели

 / tan(x)\
 \e      /
x         

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

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

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

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Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

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

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