Производная (sin(x))^(x+1)

Вы ввели

   x + 1   
sin     (x)

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

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

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

2. Теперь упростим:

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

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Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

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

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