Производная (cos(x))^(x^2)

Вы ввели

        / 2\
        \x /
(cos(x))    

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

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

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

Вторая производная

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

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

Третья производная

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

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