Производная cos(x)^x

Вы ввели

   x   
cos (x)

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

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

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

Вторая производная

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

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

Третья производная

        /                              3        2                                  /                    2   \                       3   \
   x    |     /               x*sin(x)\    3*sin (x)     /               x*sin(x)\ |    2*sin(x)   x*sin (x)|   2*x*sin(x)   2*x*sin (x)|
cos (x)*|-3 - |-log(cos(x)) + --------|  - --------- + 3*|-log(cos(x)) + --------|*|x + -------- + ---------| - ---------- - -----------|
        |     \                cos(x) /        2         \                cos(x) / |     cos(x)        2    |     cos(x)          3     |
        \                                   cos (x)                                \                cos (x) /                  cos (x)  /

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