Производная (cos(x))^atan(x)

Вы ввели

   atan(x)   
cos       (x)

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

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

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

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

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

---

Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

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

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

График