Производная (cos(4*x))^x

Вы ввели

   x     
cos (4*x)

$$\cos^{x}{\left (4 x \right )}$$

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

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

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Ответ:

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

Первая производная

   x      /  4*x*sin(4*x)                \
cos (4*x)*|- ------------ + log(cos(4*x))|
          \    cos(4*x)                  /

$$\left(- \frac{4 x \sin{\left (4 x \right )}}{\cos{\left (4 x \right )}} + \log{\left (\cos{\left (4 x \right )} \right )}\right) \cos^{x}{\left (4 x \right )}$$

Вторая производная

          /                               2                               2     \
   x      |/                 4*x*sin(4*x)\           8*sin(4*x)   16*x*sin (4*x)|
cos (4*x)*||-log(cos(4*x)) + ------------|  - 16*x - ---------- - --------------|
          |\                   cos(4*x)  /            cos(4*x)         2        |
          \                                                         cos (4*x)   /

$$\left(- \frac{16 x \sin^{2}{\left (4 x \right )}}{\cos^{2}{\left (4 x \right )}} - 16 x + \left(\frac{4 x \sin{\left (4 x \right )}}{\cos{\left (4 x \right )}} - \log{\left (\cos{\left (4 x \right )} \right )}\right)^{2} - \frac{8 \sin{\left (4 x \right )}}{\cos{\left (4 x \right )}}\right) \cos^{x}{\left (4 x \right )}$$

Третья производная

          /                                     3         2                                           /                        2     \                             3     \
   x      |      /                 4*x*sin(4*x)\    48*sin (4*x)      /                 4*x*sin(4*x)\ |      sin(4*x)   2*x*sin (4*x)|   128*x*sin(4*x)   128*x*sin (4*x)|
cos (4*x)*|-48 - |-log(cos(4*x)) + ------------|  - ------------ + 24*|-log(cos(4*x)) + ------------|*|2*x + -------- + -------------| - -------------- - ---------------|
          |      \                   cos(4*x)  /        2             \                   cos(4*x)  / |      cos(4*x)        2       |      cos(4*x)            3        |
          \                                          cos (4*x)                                        \                   cos (4*x)  /                       cos (4*x)   /

$$\left(- \frac{128 x \sin^{3}{\left (4 x \right )}}{\cos^{3}{\left (4 x \right )}} - \frac{128 x \sin{\left (4 x \right )}}{\cos{\left (4 x \right )}} - \left(\frac{4 x \sin{\left (4 x \right )}}{\cos{\left (4 x \right )}} - \log{\left (\cos{\left (4 x \right )} \right )}\right)^{3} + 24 \left(\frac{4 x \sin{\left (4 x \right )}}{\cos{\left (4 x \right )}} - \log{\left (\cos{\left (4 x \right )} \right )}\right) \left(\frac{2 x \sin^{2}{\left (4 x \right )}}{\cos^{2}{\left (4 x \right )}} + 2 x + \frac{\sin{\left (4 x \right )}}{\cos{\left (4 x \right )}}\right) - \frac{48 \sin^{2}{\left (4 x \right )}}{\cos^{2}{\left (4 x \right )}} - 48\right) \cos^{x}{\left (4 x \right )}$$