Производная tan(x)^(cos(x))

Вы ввели

   cos(x)   
tan      (x)

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

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

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

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Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

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

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