Производная (sin(x)+x^2)^(cos(x))

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↑ Функция f () ? - производная -го порядка

Решение

Вы ввели
[LaTeX]
             cos(x)
/          2\      
\sin(x) + x /      
$$\left(x^{2} + \sin{\left (x \right )}\right)^{\cos{\left (x \right )}}$$
Подробное решение
[LaTeX]
  1. Не могу найти шаги в поиске этой производной.

    Но производная


Ответ:

График
[LaTeX]
Первая производная
[LaTeX]
             cos(x)                                                    
/          2\       /     /          2\          (2*x + cos(x))*cos(x)\
\sin(x) + x /      *|- log\sin(x) + x /*sin(x) + ---------------------|
                    |                                           2     |
                    \                                 sin(x) + x      /
$$\left(x^{2} + \sin{\left (x \right )}\right)^{\cos{\left (x \right )}} \left(\frac{\left(2 x + \cos{\left (x \right )}\right) \cos{\left (x \right )}}{x^{2} + \sin{\left (x \right )}} - \log{\left (x^{2} + \sin{\left (x \right )} \right )} \sin{\left (x \right )}\right)$$
Вторая производная
[LaTeX]
             cos(x) /                                                   2                                                                  2                                 \
/ 2         \       |/     / 2         \          (2*x + cos(x))*cos(x)\              / 2         \   (-2 + sin(x))*cos(x)   (2*x + cos(x)) *cos(x)   2*(2*x + cos(x))*sin(x)|
\x  + sin(x)/      *||- log\x  + sin(x)/*sin(x) + ---------------------|  - cos(x)*log\x  + sin(x)/ - -------------------- - ---------------------- - -----------------------|
                    ||                                  2              |                                   2                                  2              2               |
                    |\                                 x  + sin(x)     /                                  x  + sin(x)            / 2         \              x  + sin(x)      |
                    \                                                                                                            \x  + sin(x)/                               /
$$\left(x^{2} + \sin{\left (x \right )}\right)^{\cos{\left (x \right )}} \left(- \frac{\left(2 x + \cos{\left (x \right )}\right)^{2} \cos{\left (x \right )}}{\left(x^{2} + \sin{\left (x \right )}\right)^{2}} - \frac{2 \left(2 x + \cos{\left (x \right )}\right) \sin{\left (x \right )}}{x^{2} + \sin{\left (x \right )}} + \left(\frac{\left(2 x + \cos{\left (x \right )}\right) \cos{\left (x \right )}}{x^{2} + \sin{\left (x \right )}} - \log{\left (x^{2} + \sin{\left (x \right )} \right )} \sin{\left (x \right )}\right)^{2} - \log{\left (x^{2} + \sin{\left (x \right )} \right )} \cos{\left (x \right )} - \frac{\left(\sin{\left (x \right )} - 2\right) \cos{\left (x \right )}}{x^{2} + \sin{\left (x \right )}}\right)$$
Третья производная
[LaTeX]
             cos(x) /                                                   3                                  2                                                              /                                                               2                                 \                                             3                                                   2                                               \
/ 2         \       |/     / 2         \          (2*x + cos(x))*cos(x)\       / 2         \            cos (x)       /     / 2         \          (2*x + cos(x))*cos(x)\ |          / 2         \   (-2 + sin(x))*cos(x)   (2*x + cos(x)) *cos(x)   2*(2*x + cos(x))*sin(x)|   3*(2*x + cos(x))*cos(x)   2*(2*x + cos(x)) *cos(x)   3*(-2 + sin(x))*sin(x)   3*(2*x + cos(x)) *sin(x)   3*(-2 + sin(x))*(2*x + cos(x))*cos(x)|
\x  + sin(x)/      *||- log\x  + sin(x)/*sin(x) + ---------------------|  + log\x  + sin(x)/*sin(x) - ----------- - 3*|- log\x  + sin(x)/*sin(x) + ---------------------|*|cos(x)*log\x  + sin(x)/ + -------------------- + ---------------------- + -----------------------| - ----------------------- + ------------------------ + ---------------------- + ------------------------ + -------------------------------------|
                    ||                                  2              |                               2              |                                  2              | |                               2                                  2              2               |          2                                    3              2                                    2                                 2           |
                    |\                                 x  + sin(x)     /                              x  + sin(x)     \                                 x  + sin(x)     / |                              x  + sin(x)            / 2         \              x  + sin(x)      |         x  + sin(x)              / 2         \              x  + sin(x)              / 2         \                     / 2         \            |
                    \                                                                                                                                                     \                                                     \x  + sin(x)/                               /                                  \x  + sin(x)/                                       \x  + sin(x)/                     \x  + sin(x)/            /
$$\left(x^{2} + \sin{\left (x \right )}\right)^{\cos{\left (x \right )}} \left(\frac{2 \left(2 x + \cos{\left (x \right )}\right)^{3} \cos{\left (x \right )}}{\left(x^{2} + \sin{\left (x \right )}\right)^{3}} + \frac{3 \left(2 x + \cos{\left (x \right )}\right)^{2} \sin{\left (x \right )}}{\left(x^{2} + \sin{\left (x \right )}\right)^{2}} - \frac{3 \left(2 x + \cos{\left (x \right )}\right) \cos{\left (x \right )}}{x^{2} + \sin{\left (x \right )}} + \frac{3 \cos{\left (x \right )}}{\left(x^{2} + \sin{\left (x \right )}\right)^{2}} \left(2 x + \cos{\left (x \right )}\right) \left(\sin{\left (x \right )} - 2\right) + \left(\frac{\left(2 x + \cos{\left (x \right )}\right) \cos{\left (x \right )}}{x^{2} + \sin{\left (x \right )}} - \log{\left (x^{2} + \sin{\left (x \right )} \right )} \sin{\left (x \right )}\right)^{3} - 3 \left(\frac{\left(2 x + \cos{\left (x \right )}\right) \cos{\left (x \right )}}{x^{2} + \sin{\left (x \right )}} - \log{\left (x^{2} + \sin{\left (x \right )} \right )} \sin{\left (x \right )}\right) \left(\frac{\left(2 x + \cos{\left (x \right )}\right)^{2} \cos{\left (x \right )}}{\left(x^{2} + \sin{\left (x \right )}\right)^{2}} + \frac{2 \left(2 x + \cos{\left (x \right )}\right) \sin{\left (x \right )}}{x^{2} + \sin{\left (x \right )}} + \log{\left (x^{2} + \sin{\left (x \right )} \right )} \cos{\left (x \right )} + \frac{\left(\sin{\left (x \right )} - 2\right) \cos{\left (x \right )}}{x^{2} + \sin{\left (x \right )}}\right) + \log{\left (x^{2} + \sin{\left (x \right )} \right )} \sin{\left (x \right )} + \frac{3 \left(\sin{\left (x \right )} - 2\right) \sin{\left (x \right )}}{x^{2} + \sin{\left (x \right )}} - \frac{\cos^{2}{\left (x \right )}}{x^{2} + \sin{\left (x \right )}}\right)$$