Производная x^(sin(x+1))

Вы ввели

 sin(x + 1)
x          

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

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

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

2. Теперь упростим:

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

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Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

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

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