Производная x^tan(x)

Вы ввели

 tan(x)
x      

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

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

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

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

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

Вторая производная

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

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

Третья производная

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

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

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