Производная sin(x)^x

Вы ввели

   x   
sin (x)

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

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

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

Вторая производная

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

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

Третья производная

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

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