Производная (1/x)^tan(x)

Вы ввели

   tan(x)
/1\      
|-|      
\x/      

$$\left(\frac{1}{x}\right)^{\tan{\left (x \right )}}$$

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

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

---

Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

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

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