Производная (cos(x))^(x-3)

Вы ввели

   x - 3   
cos     (x)

$$\cos^{x - 3}{\left (x \right )}$$

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

   $\left(x - 3\right)^{x - 3} \left(\log{\left (x - 3 \right )} + 1\right)$

2. Теперь упростим:

   $\left(x - 3\right)^{x - 3} \left(\log{\left (x - 3 \right )} + 1\right)$

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Ответ:

$\left(x - 3\right)^{x - 3} \left(\log{\left (x - 3 \right )} + 1\right)$

Первая производная

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

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

Вторая производная

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

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

Третья производная

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

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