Производная log(x)^tan(x)

Вы ввели

   tan(x)   
log      (x)

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

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

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

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Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

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

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