Производная (1+cos(x))^sec(x)

Вы ввели

            sec(x)
(1 + cos(x))      

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

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

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

---

Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

                   /                                                  3                                            3             2               /       2   \                                 2                                                                       /                   2                                                                                  \                                                \       
            sec(x) |  sin(x)     /                           sin(x)  \     2         3                        2*sin (x)     3*tan (x)*sin(x)   3*\1 + tan (x)/*sin(x)   3*cos(x)*tan(x)   3*sin (x)*tan(x)   3*cos(x)*sin(x)     /                           sin(x)  \ |  cos(x)        sin (x)         2                      /       2   \                   2*sin(x)*tan(x)|            /       2   \                       |       
(1 + cos(x))      *|---------- + |log(1 + cos(x))*tan(x) - ----------| *sec (x) + tan (x)*log(1 + cos(x)) - ------------- - ---------------- - ---------------------- - --------------- - ---------------- - --------------- - 3*|log(1 + cos(x))*tan(x) - ----------|*|---------- + ------------- - tan (x)*log(1 + cos(x)) - \1 + tan (x)/*log(1 + cos(x)) + ---------------|*sec(x) + 5*\1 + tan (x)/*log(1 + cos(x))*tan(x)|*sec(x)
                   |1 + cos(x)   \                         1 + cos(x)/                                                  3      1 + cos(x)            1 + cos(x)            1 + cos(x)                  2                  2      \                         1 + cos(x)/ |1 + cos(x)               2                                                                1 + cos(x)  |                                                |       
                   \                                                                                        (1 + cos(x))                                                                   (1 + cos(x))       (1 + cos(x))                                             \             (1 + cos(x))                                                                             /                                                /       

$$\left(\cos{\left (x \right )} + 1\right)^{\sec{\left (x \right )}} \left(\left(\log{\left (\cos{\left (x \right )} + 1 \right )} \tan{\left (x \right )} - \frac{\sin{\left (x \right )}}{\cos{\left (x \right )} + 1}\right)^{3} \sec^{2}{\left (x \right )} - 3 \left(\log{\left (\cos{\left (x \right )} + 1 \right )} \tan{\left (x \right )} - \frac{\sin{\left (x \right )}}{\cos{\left (x \right )} + 1}\right) \left(- \left(\tan^{2}{\left (x \right )} + 1\right) \log{\left (\cos{\left (x \right )} + 1 \right )} - \log{\left (\cos{\left (x \right )} + 1 \right )} \tan^{2}{\left (x \right )} + \frac{2 \sin{\left (x \right )} \tan{\left (x \right )}}{\cos{\left (x \right )} + 1} + \frac{\cos{\left (x \right )}}{\cos{\left (x \right )} + 1} + \frac{\sin^{2}{\left (x \right )}}{\left(\cos{\left (x \right )} + 1\right)^{2}}\right) \sec{\left (x \right )} + 5 \left(\tan^{2}{\left (x \right )} + 1\right) \log{\left (\cos{\left (x \right )} + 1 \right )} \tan{\left (x \right )} + \log{\left (\cos{\left (x \right )} + 1 \right )} \tan^{3}{\left (x \right )} - \frac{3 \left(\tan^{2}{\left (x \right )} + 1\right) \sin{\left (x \right )}}{\cos{\left (x \right )} + 1} - \frac{3 \sin{\left (x \right )} \tan^{2}{\left (x \right )}}{\cos{\left (x \right )} + 1} + \frac{\sin{\left (x \right )}}{\cos{\left (x \right )} + 1} - \frac{3 \cos{\left (x \right )} \tan{\left (x \right )}}{\cos{\left (x \right )} + 1} - \frac{3 \sin^{2}{\left (x \right )} \tan{\left (x \right )}}{\left(\cos{\left (x \right )} + 1\right)^{2}} - \frac{3 \sin{\left (x \right )} \cos{\left (x \right )}}{\left(\cos{\left (x \right )} + 1\right)^{2}} - \frac{2 \sin^{3}{\left (x \right )}}{\left(\cos{\left (x \right )} + 1\right)^{3}}\right) \sec{\left (x \right )}$$