Производная sin(x)^tan(x)

Вы ввели

   tan(x)   
sin      (x)

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

d /   tan(x)   \
--\sin      (x)/
dx              

$$\frac{d}{d x} \sin^{\tan{\left(x \right)}}{\left(x \right)}$$

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

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

Вторая производная

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

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

Третья производная

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

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

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