Производная x^atan(7*x)

Вы ввели

 atan(7*x)
x         

$$x^{\operatorname{atan}{\left (7 x \right )}}$$

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

   $\left(\log{\left (\operatorname{atan}{\left (7 x \right )} \right )} + 1\right) \operatorname{atan}^{\operatorname{atan}{\left (7 x \right )}}{\left (7 x \right )}$

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Ответ:

$\left(\log{\left (\operatorname{atan}{\left (7 x \right )} \right )} + 1\right) \operatorname{atan}^{\operatorname{atan}{\left (7 x \right )}}{\left (7 x \right )}$

Первая производная

 atan(7*x) /atan(7*x)    7*log(x)\
x         *|--------- + ---------|
           |    x               2|
           \            1 + 49*x /

$$x^{\operatorname{atan}{\left (7 x \right )}} \left(\frac{7 \log{\left (x \right )}}{49 x^{2} + 1} + \frac{1}{x} \operatorname{atan}{\left (7 x \right )}\right)$$

Вторая производная

           /                       2                                           \
 atan(7*x) |/atan(7*x)    7*log(x)\    atan(7*x)         14        686*x*log(x)|
x         *||--------- + ---------|  - --------- + ------------- - ------------|
           ||    x               2|         2        /        2\              2|
           |\            1 + 49*x /        x       x*\1 + 49*x /   /        2\ |
           \                                                       \1 + 49*x / /

$$x^{\operatorname{atan}{\left (7 x \right )}} \left(- \frac{686 x \log{\left (x \right )}}{\left(49 x^{2} + 1\right)^{2}} + \left(\frac{7 \log{\left (x \right )}}{49 x^{2} + 1} + \frac{1}{x} \operatorname{atan}{\left (7 x \right )}\right)^{2} + \frac{14}{x \left(49 x^{2} + 1\right)} - \frac{1}{x^{2}} \operatorname{atan}{\left (7 x \right )}\right)$$

Третья производная

           /                       3                                                                                                                                               2       \
 atan(7*x) |/atan(7*x)    7*log(x)\        2058        686*log(x)          21           /atan(7*x)    7*log(x)\ /atan(7*x)         14        686*x*log(x)\   2*atan(7*x)   134456*x *log(x)|
x         *||--------- + ---------|  - ------------ - ------------ - -------------- - 3*|--------- + ---------|*|--------- - ------------- + ------------| + ----------- + ----------------|
           ||    x               2|               2              2    2 /        2\     |    x               2| |     2        /        2\              2|         3                    3  |
           |\            1 + 49*x /    /        2\    /        2\    x *\1 + 49*x /     \            1 + 49*x / |    x       x*\1 + 49*x /   /        2\ |        x          /        2\   |
           \                           \1 + 49*x /    \1 + 49*x /                                               \                            \1 + 49*x / /                   \1 + 49*x /   /

$$x^{\operatorname{atan}{\left (7 x \right )}} \left(\frac{134456 x^{2} \log{\left (x \right )}}{\left(49 x^{2} + 1\right)^{3}} + \left(\frac{7 \log{\left (x \right )}}{49 x^{2} + 1} + \frac{1}{x} \operatorname{atan}{\left (7 x \right )}\right)^{3} - 3 \left(\frac{7 \log{\left (x \right )}}{49 x^{2} + 1} + \frac{1}{x} \operatorname{atan}{\left (7 x \right )}\right) \left(\frac{686 x \log{\left (x \right )}}{\left(49 x^{2} + 1\right)^{2}} - \frac{14}{x \left(49 x^{2} + 1\right)} + \frac{1}{x^{2}} \operatorname{atan}{\left (7 x \right )}\right) - \frac{686 \log{\left (x \right )}}{\left(49 x^{2} + 1\right)^{2}} - \frac{2058}{\left(49 x^{2} + 1\right)^{2}} - \frac{21}{x^{2} \left(49 x^{2} + 1\right)} + \frac{2}{x^{3}} \operatorname{atan}{\left (7 x \right )}\right)$$