Производная (atan(x))^sin(x)

Вы ввели

    sin(x)   
atan      (x)

$$\operatorname{atan}^{\sin{\left(x \right)}}{\left(x \right)}$$

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

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

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

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

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

Вторая производная

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

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

Третья производная

              /                                        3                                                                                                                                                                                                                                                                                        2          \
    sin(x)    |/                           sin(x)     \                            /                           sin(x)     \ /                            sin(x)             2*cos(x)           2*x*sin(x)   \       3*sin(x)            3*cos(x)             2*sin(x)            2*sin(x)            6*x*cos(x)          6*x*sin(x)          8*x *sin(x)   |
atan      (x)*||cos(x)*log(atan(x)) + ----------------|  - cos(x)*log(atan(x)) - 3*|cos(x)*log(atan(x)) + ----------------|*|log(atan(x))*sin(x) + ------------------ - ---------------- + -----------------| - ---------------- - ------------------ - ----------------- + ------------------ - ----------------- + ------------------ + -----------------|
              ||                      /     2\        |                            |                      /     2\        | |                              2            /     2\                   2        |   /     2\                   2                    2                   3                    2                   3                    3        |
              |\                      \1 + x /*atan(x)/                            \                      \1 + x /*atan(x)/ |                      /     2\      2      \1 + x /*atan(x)   /     2\         |   \1 + x /*atan(x)   /     2\      2      /     2\            /     2\      3      /     2\            /     2\      2      /     2\         |
              \                                                                                                             \                      \1 + x / *atan (x)                      \1 + x / *atan(x)/                      \1 + x / *atan (x)   \1 + x / *atan(x)   \1 + x / *atan (x)   \1 + x / *atan(x)   \1 + x / *atan (x)   \1 + x / *atan(x)/

$$\left(\frac{8 x^{2} \sin{\left(x \right)}}{\left(x^{2} + 1\right)^{3} \operatorname{atan}{\left(x \right)}} - \frac{6 x \cos{\left(x \right)}}{\left(x^{2} + 1\right)^{2} \operatorname{atan}{\left(x \right)}} + \frac{6 x \sin{\left(x \right)}}{\left(x^{2} + 1\right)^{3} \operatorname{atan}^{2}{\left(x \right)}} + \left(\log{\left(\operatorname{atan}{\left(x \right)} \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{\left(x^{2} + 1\right) \operatorname{atan}{\left(x \right)}}\right)^{3} - 3 \left(\log{\left(\operatorname{atan}{\left(x \right)} \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{\left(x^{2} + 1\right) \operatorname{atan}{\left(x \right)}}\right) \left(\frac{2 x \sin{\left(x \right)}}{\left(x^{2} + 1\right)^{2} \operatorname{atan}{\left(x \right)}} + \log{\left(\operatorname{atan}{\left(x \right)} \right)} \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)}}{\left(x^{2} + 1\right) \operatorname{atan}{\left(x \right)}} + \frac{\sin{\left(x \right)}}{\left(x^{2} + 1\right)^{2} \operatorname{atan}^{2}{\left(x \right)}}\right) - \log{\left(\operatorname{atan}{\left(x \right)} \right)} \cos{\left(x \right)} - \frac{3 \sin{\left(x \right)}}{\left(x^{2} + 1\right) \operatorname{atan}{\left(x \right)}} - \frac{2 \sin{\left(x \right)}}{\left(x^{2} + 1\right)^{2} \operatorname{atan}{\left(x \right)}} - \frac{3 \cos{\left(x \right)}}{\left(x^{2} + 1\right)^{2} \operatorname{atan}^{2}{\left(x \right)}} + \frac{2 \sin{\left(x \right)}}{\left(x^{2} + 1\right)^{3} \operatorname{atan}^{3}{\left(x \right)}}\right) \operatorname{atan}^{\sin{\left(x \right)}}{\left(x \right)}$$

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