Производная x^atan(x)^(2)

Вы ввели

     2   
 atan (x)
x        

$$x^{\operatorname{atan}^{2}{\left (x \right )}}$$

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

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

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Ответ:

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

Первая производная

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

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

Вторая производная

     2    /                    2                2                                                 \
 atan (x) |/atan(x)   2*log(x)\      2      atan (x)    2*log(x)   4*atan(x)    4*x*atan(x)*log(x)|
x        *||------- + --------| *atan (x) - -------- + --------- + ---------- - ------------------|
          ||   x            2 |                 2              2     /     2\               2     |
          |\           1 + x  /                x       /     2\    x*\1 + x /       /     2\      |
          \                                            \1 + x /                     \1 + x /      /

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

Третья производная

     2    /                    3                               2                                                                                          /    2                                                 \               2               \
 atan (x) |/atan(x)   2*log(x)\      3      12*atan(x)   2*atan (x)        6        12*x*log(x)    6*atan(x)    4*atan(x)*log(x)     /atan(x)   2*log(x)\ |atan (x)    2*log(x)   4*atan(x)    4*x*atan(x)*log(x)|           16*x *atan(x)*log(x)|
x        *||------- + --------| *atan (x) - ---------- + ---------- + ----------- - ----------- - ----------- - ---------------- - 3*|------- + --------|*|-------- - --------- - ---------- + ------------------|*atan(x) + --------------------|
          ||   x            2 |                     2         3                 2            3     2 /     2\              2         |   x            2 | |    2              2     /     2\               2     |                        3      |
          |\           1 + x  /             /     2\         x          /     2\     /     2\     x *\1 + x /      /     2\          \           1 + x  / |   x       /     2\    x*\1 + x /       /     2\      |                /     2\       |
          \                                 \1 + x /                  x*\1 + x /     \1 + x /                      \1 + x /                               \           \1 + x /                     \1 + x /      /                \1 + x /       /

$$x^{\operatorname{atan}^{2}{\left (x \right )}} \left(\frac{16 x^{2} \operatorname{atan}{\left (x \right )}}{\left(x^{2} + 1\right)^{3}} \log{\left (x \right )} - \frac{12 x \log{\left (x \right )}}{\left(x^{2} + 1\right)^{3}} + \left(\frac{2 \log{\left (x \right )}}{x^{2} + 1} + \frac{1}{x} \operatorname{atan}{\left (x \right )}\right)^{3} \operatorname{atan}^{3}{\left (x \right )} - 3 \left(\frac{2 \log{\left (x \right )}}{x^{2} + 1} + \frac{1}{x} \operatorname{atan}{\left (x \right )}\right) \left(\frac{4 x \operatorname{atan}{\left (x \right )}}{\left(x^{2} + 1\right)^{2}} \log{\left (x \right )} - \frac{2 \log{\left (x \right )}}{\left(x^{2} + 1\right)^{2}} - \frac{4 \operatorname{atan}{\left (x \right )}}{x \left(x^{2} + 1\right)} + \frac{1}{x^{2}} \operatorname{atan}^{2}{\left (x \right )}\right) \operatorname{atan}{\left (x \right )} - \frac{4 \operatorname{atan}{\left (x \right )}}{\left(x^{2} + 1\right)^{2}} \log{\left (x \right )} - \frac{12 \operatorname{atan}{\left (x \right )}}{\left(x^{2} + 1\right)^{2}} + \frac{6}{x \left(x^{2} + 1\right)^{2}} - \frac{6 \operatorname{atan}{\left (x \right )}}{x^{2} \left(x^{2} + 1\right)} + \frac{2}{x^{3}} \operatorname{atan}^{2}{\left (x \right )}\right)$$