Производная atan(x)^cos(x)

Вы ввели

    cos(x)   
atan      (x)

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

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

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

---

Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

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

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

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