Производная (sin(x))^(4*x)

Вы ввели

   4*x   
sin   (x)

$$\sin^{4 x}{\left (x \right )}$$

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

   $\left(4 x\right)^{4 x} \left(\log{\left (4 x \right )} + 1\right)$

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Ответ:

$\left(4 x\right)^{4 x} \left(\log{\left (4 x \right )} + 1\right)$

Первая производная

   4*x    /                4*x*cos(x)\
sin   (x)*|4*log(sin(x)) + ----------|
          \                  sin(x)  /

$$\left(\frac{4 x \cos{\left (x \right )}}{\sin{\left (x \right )}} + 4 \log{\left (\sin{\left (x \right )} \right )}\right) \sin^{4 x}{\left (x \right )}$$

Вторая производная

            /                               2                   2   \
     4*x    |       /x*cos(x)              \    2*cos(x)   x*cos (x)|
4*sin   (x)*|-x + 4*|-------- + log(sin(x))|  + -------- - ---------|
            |       \ sin(x)               /     sin(x)        2    |
            \                                               sin (x) /

$$4 \left(- x - \frac{x \cos^{2}{\left (x \right )}}{\sin^{2}{\left (x \right )}} + 4 \left(\frac{x \cos{\left (x \right )}}{\sin{\left (x \right )}} + \log{\left (\sin{\left (x \right )} \right )}\right)^{2} + \frac{2 \cos{\left (x \right )}}{\sin{\left (x \right )}}\right) \sin^{4 x}{\left (x \right )}$$

Третья производная

            /                                3                               /                    2   \        2             3                \
     4*x    |        /x*cos(x)              \       /x*cos(x)              \ |    2*cos(x)   x*cos (x)|   3*cos (x)   2*x*cos (x)   2*x*cos(x)|
4*sin   (x)*|-3 + 16*|-------- + log(sin(x))|  - 12*|-------- + log(sin(x))|*|x - -------- + ---------| - --------- + ----------- + ----------|
            |        \ sin(x)               /       \ sin(x)               / |     sin(x)        2    |       2            3          sin(x)  |
            \                                                                \                sin (x) /    sin (x)      sin (x)               /

$$4 \left(\frac{2 x \cos{\left (x \right )}}{\sin{\left (x \right )}} + \frac{2 x \cos^{3}{\left (x \right )}}{\sin^{3}{\left (x \right )}} + 16 \left(\frac{x \cos{\left (x \right )}}{\sin{\left (x \right )}} + \log{\left (\sin{\left (x \right )} \right )}\right)^{3} - 12 \left(\frac{x \cos{\left (x \right )}}{\sin{\left (x \right )}} + \log{\left (\sin{\left (x \right )} \right )}\right) \left(x + \frac{x \cos^{2}{\left (x \right )}}{\sin^{2}{\left (x \right )}} - \frac{2 \cos{\left (x \right )}}{\sin{\left (x \right )}}\right) - 3 - \frac{3 \cos^{2}{\left (x \right )}}{\sin^{2}{\left (x \right )}}\right) \sin^{4 x}{\left (x \right )}$$