Производная (1+x)^cos(x)

Вы ввели

       cos(x)
(1 + x)      

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

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

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

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Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

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

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