Производная (tan(x))^sin(x)

Вы ввели

   sin(x)   
tan      (x)

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

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

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

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

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

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Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

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

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

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