Производная sin(x)^x^2

Вы ввели

        / 2\
        \x /
(sin(x))    

$$\sin^{x^{2}}{\left (x \right )}$$

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

   $\left(\log{\left (x^{2} \right )} + 1\right) \left(x^{2}\right)^{x^{2}}$

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Ответ:

$\left(\log{\left (x^{2} \right )} + 1\right) \left(x^{2}\right)^{x^{2}}$

Первая производная

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

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

Вторая производная

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

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

Третья производная

        / 2\ /                                    3                     2                                     /                      2    2                \      2    3         2       \
        \x / |        3 /                x*cos(x)\    6*cos(x)   6*x*cos (x)       /                x*cos(x)\ | 2                   x *cos (x)   4*x*cos(x)|   2*x *cos (x)   2*x *cos(x)|
(sin(x))    *|-6*x + x *|2*log(sin(x)) + --------|  + -------- - ----------- - 3*x*|2*log(sin(x)) + --------|*|x  - 2*log(sin(x)) + ---------- - ----------| + ------------ + -----------|
             |          \                 sin(x) /     sin(x)         2            \                 sin(x) / |                         2          sin(x)  |        3            sin(x)  |
             \                                                     sin (x)                                    \                      sin (x)               /     sin (x)                 /

$$\left(x^{3} \left(\frac{x \cos{\left (x \right )}}{\sin{\left (x \right )}} + 2 \log{\left (\sin{\left (x \right )} \right )}\right)^{3} + \frac{2 x^{2} \cos{\left (x \right )}}{\sin{\left (x \right )}} + \frac{2 x^{2} \cos^{3}{\left (x \right )}}{\sin^{3}{\left (x \right )}} - 3 x \left(\frac{x \cos{\left (x \right )}}{\sin{\left (x \right )}} + 2 \log{\left (\sin{\left (x \right )} \right )}\right) \left(x^{2} + \frac{x^{2} \cos^{2}{\left (x \right )}}{\sin^{2}{\left (x \right )}} - \frac{4 x \cos{\left (x \right )}}{\sin{\left (x \right )}} - 2 \log{\left (\sin{\left (x \right )} \right )}\right) - 6 x - \frac{6 x \cos^{2}{\left (x \right )}}{\sin^{2}{\left (x \right )}} + \frac{6 \cos{\left (x \right )}}{\sin{\left (x \right )}}\right) \sin^{x^{2}}{\left (x \right )}$$