Производная x^tan(x)^3

Вы ввели

    3   
 tan (x)
x       

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

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

   $\left(\log{\left (\tan^{3}{\left (x \right )} \right )} + 1\right) \left(\tan^{3}{\left (x \right )}\right)^{\tan^{3}{\left (x \right )}}$

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Ответ:

$\left(\log{\left (\tan^{3}{\left (x \right )} \right )} + 1\right) \left(\tan^{3}{\left (x \right )}\right)^{\tan^{3}{\left (x \right )}}$

Первая производная

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

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

Вторая производная

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

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

Третья производная

         /                                                                                                                                                                                                                                                                                                             2                                                                     \
    3    |                                 3                3                     3               2    /       2   \                                               /     2                     2            /       2   \                                        \                                        /       2   \                 3    /       2   \                   2               |
 tan (x) |/tan(x)     /       2   \       \     6      2*tan (x)     /       2   \           9*tan (x)*\1 + tan (x)/        3    /tan(x)     /       2   \       \ |  tan (x)     /       2   \           6*\1 + tan (x)/*tan(x)        2    /       2   \       |         4    /       2   \          18*\1 + tan (x)/ *tan(x)   18*tan (x)*\1 + tan (x)/      /       2   \     2          |
x       *||------ + 3*\1 + tan (x)/*log(x)| *tan (x) + --------- + 6*\1 + tan (x)/ *log(x) - ----------------------- + 3*tan (x)*|------ + 3*\1 + tan (x)/*log(x)|*|- ------- + 6*\1 + tan (x)/ *log(x) + ---------------------- + 6*tan (x)*\1 + tan (x)/*log(x)| + 12*tan (x)*\1 + tan (x)/*log(x) + ------------------------ + ------------------------ + 42*\1 + tan (x)/ *tan (x)*log(x)|
         |\  x                            /                 3                                            2                       \  x                            / |      2                                         x                                            |                                                x                          x                                               |
         \                                                 x                                            x                                                          \     x                                                                                       /                                                                                                                           /

$$x^{\tan^{3}{\left (x \right )}} \left(\left(3 \left(\tan^{2}{\left (x \right )} + 1\right) \log{\left (x \right )} + \frac{1}{x} \tan{\left (x \right )}\right)^{3} \tan^{6}{\left (x \right )} + 3 \left(3 \left(\tan^{2}{\left (x \right )} + 1\right) \log{\left (x \right )} + \frac{1}{x} \tan{\left (x \right )}\right) \left(6 \left(\tan^{2}{\left (x \right )} + 1\right)^{2} \log{\left (x \right )} + 6 \left(\tan^{2}{\left (x \right )} + 1\right) \log{\left (x \right )} \tan^{2}{\left (x \right )} + \frac{6}{x} \left(\tan^{2}{\left (x \right )} + 1\right) \tan{\left (x \right )} - \frac{1}{x^{2}} \tan^{2}{\left (x \right )}\right) \tan^{3}{\left (x \right )} + 6 \left(\tan^{2}{\left (x \right )} + 1\right)^{3} \log{\left (x \right )} + 42 \left(\tan^{2}{\left (x \right )} + 1\right)^{2} \log{\left (x \right )} \tan^{2}{\left (x \right )} + 12 \left(\tan^{2}{\left (x \right )} + 1\right) \log{\left (x \right )} \tan^{4}{\left (x \right )} + \frac{18}{x} \left(\tan^{2}{\left (x \right )} + 1\right)^{2} \tan{\left (x \right )} + \frac{18}{x} \left(\tan^{2}{\left (x \right )} + 1\right) \tan^{3}{\left (x \right )} - \frac{9}{x^{2}} \left(\tan^{2}{\left (x \right )} + 1\right) \tan^{2}{\left (x \right )} + \frac{2}{x^{3}} \tan^{3}{\left (x \right )}\right)$$