Производная x^sin(x)^2

Вы ввели

    2   
 sin (x)
x       

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

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

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

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Ответ:

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

Первая производная

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

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

Вторая производная

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

$$x^{\sin^{2}{\left (x \right )}} \left(\left(2 \log{\left (x \right )} \cos{\left (x \right )} + \frac{1}{x} \sin{\left (x \right )}\right)^{2} \sin^{2}{\left (x \right )} - 2 \log{\left (x \right )} \sin^{2}{\left (x \right )} + 2 \log{\left (x \right )} \cos^{2}{\left (x \right )} + \frac{4}{x} \sin{\left (x \right )} \cos{\left (x \right )} - \frac{1}{x^{2}} \sin^{2}{\left (x \right )}\right)$$

Третья производная

    2    /                          3                2           2           2                                                                              /   2                                                           \       \
 sin (x) |/sin(x)                  \     3      6*sin (x)   2*sin (x)   6*cos (x)                            6*cos(x)*sin(x)     /sin(x)                  \ |sin (x)        2                  2             4*cos(x)*sin(x)|       |
x       *||------ + 2*cos(x)*log(x)| *sin (x) - --------- + --------- + --------- - 8*cos(x)*log(x)*sin(x) - --------------- - 3*|------ + 2*cos(x)*log(x)|*|------- - 2*cos (x)*log(x) + 2*sin (x)*log(x) - ---------------|*sin(x)|
         |\  x                     /                x            3          x                                        2           \  x                     / |    2                                                  x       |       |
         \                                                      x                                                   x                                       \   x                                                           /       /

$$x^{\sin^{2}{\left (x \right )}} \left(\left(2 \log{\left (x \right )} \cos{\left (x \right )} + \frac{1}{x} \sin{\left (x \right )}\right)^{3} \sin^{3}{\left (x \right )} - 3 \left(2 \log{\left (x \right )} \cos{\left (x \right )} + \frac{1}{x} \sin{\left (x \right )}\right) \left(2 \log{\left (x \right )} \sin^{2}{\left (x \right )} - 2 \log{\left (x \right )} \cos^{2}{\left (x \right )} - \frac{4}{x} \sin{\left (x \right )} \cos{\left (x \right )} + \frac{1}{x^{2}} \sin^{2}{\left (x \right )}\right) \sin{\left (x \right )} - 8 \log{\left (x \right )} \sin{\left (x \right )} \cos{\left (x \right )} - \frac{6}{x} \sin^{2}{\left (x \right )} + \frac{6}{x} \cos^{2}{\left (x \right )} - \frac{6}{x^{2}} \sin{\left (x \right )} \cos{\left (x \right )} + \frac{2}{x^{3}} \sin^{2}{\left (x \right )}\right)$$