Производная (tan(3*x))^x

Вы ввели

   x     
tan (3*x)

$$\tan^{x}{\left (3 x \right )}$$

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

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

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Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

          /                                          3                                    2                        2                                                                                                                        3                                 \
          |     /    /       2     \                \                      /       2     \          /       2     \                      /    /       2     \                \ /                     /       2     \\        /       2     \                                  |
   x      |     |3*x*\1 + tan (3*x)/                |          2        27*\1 + tan (3*x)/    108*x*\1 + tan (3*x)/      /       2     \ |3*x*\1 + tan (3*x)/                | |   2             3*x*\1 + tan (3*x)/|   54*x*\1 + tan (3*x)/          /       2     \         |
tan (3*x)*|54 + |------------------- + log(tan(3*x))|  + 54*tan (3*x) - ------------------- - ---------------------- + 9*\1 + tan (3*x)/*|------------------- + log(tan(3*x))|*|-------- + 6*x - -------------------| + --------------------- + 108*x*\1 + tan (3*x)/*tan(3*x)|
          |     \      tan(3*x)                     /                           2                    tan(3*x)                            \      tan(3*x)                     / |tan(3*x)                 2          |            3                                            |
          \                                                                  tan (3*x)                                                                                         \                      tan (3*x)     /         tan (3*x)                                       /

$$\left(\frac{54 x \left(\tan^{2}{\left (3 x \right )} + 1\right)^{3}}{\tan^{3}{\left (3 x \right )}} - \frac{108 x \left(\tan^{2}{\left (3 x \right )} + 1\right)^{2}}{\tan{\left (3 x \right )}} + 108 x \left(\tan^{2}{\left (3 x \right )} + 1\right) \tan{\left (3 x \right )} + \left(\frac{3 x \left(\tan^{2}{\left (3 x \right )} + 1\right)}{\tan{\left (3 x \right )}} + \log{\left (\tan{\left (3 x \right )} \right )}\right)^{3} + 9 \left(\frac{3 x \left(\tan^{2}{\left (3 x \right )} + 1\right)}{\tan{\left (3 x \right )}} + \log{\left (\tan{\left (3 x \right )} \right )}\right) \left(\tan^{2}{\left (3 x \right )} + 1\right) \left(- \frac{3 x \left(\tan^{2}{\left (3 x \right )} + 1\right)}{\tan^{2}{\left (3 x \right )}} + 6 x + \frac{2}{\tan{\left (3 x \right )}}\right) - \frac{27 \left(\tan^{2}{\left (3 x \right )} + 1\right)^{2}}{\tan^{2}{\left (3 x \right )}} + 54 \tan^{2}{\left (3 x \right )} + 54\right) \tan^{x}{\left (3 x \right )}$$