Производная sin(x^atan(x))

Вы ввели

   / atan(x)\
sin\x       /

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

d /   / atan(x)\\
--\sin\x       //
dx               

$$\frac{d}{d x} \sin{\left(x^{\operatorname{atan}{\left(x \right)}} \right)}$$

Подробное решение

1. Заменим $u = x^{\operatorname{atan}{\left(x \right)}}$.

2. Производная синуса есть косинус:

   $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$

3. Затем примените цепочку правил. Умножим на $\frac{d}{d x} x^{\operatorname{atan}{\left(x \right)}}$:

   1. Не могу найти шаги в поиске этой производной.

      Но производная

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

   В результате последовательности правил:

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

---

Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

         /                  3                 /                                                     2       \                                              3                                              3                                                                                                                                                                         \
 atan(x) |/atan(x)   log(x)\     / atan(x)\   |    6       2*atan(x)    2*log(x)        3        8*x *log(x)|    / atan(x)\    2*atan(x) /atan(x)   log(x)\     / atan(x)\      atan(x) /atan(x)   log(x)\     / atan(x)\     /atan(x)   log(x)\ /atan(x)       2        2*x*log(x)\    / atan(x)\      atan(x) /atan(x)   log(x)\ /atan(x)       2        2*x*log(x)\    / atan(x)\|
x       *||------- + ------| *cos\x       / - |--------- - --------- + --------- + ----------- - -----------|*cos\x       / - x         *|------- + ------| *cos\x       / - 3*x       *|------- + ------| *sin\x       / - 3*|------- + ------|*|------- - ---------- + ----------|*cos\x       / + 3*x       *|------- + ------|*|------- - ---------- + ----------|*sin\x       /|
         ||   x           2|                  |        2        3              2    2 /     2\            3 |                            |   x           2|                             |   x           2|                    |   x           2| |    2       /     2\           2 |                            |   x           2| |    2       /     2\           2 |              |
         |\          1 + x /                  |/     2\        x       /     2\    x *\1 + x /    /     2\  |                            \          1 + x /                             \          1 + x /                    \          1 + x / |   x      x*\1 + x /   /     2\  |                            \          1 + x / |   x      x*\1 + x /   /     2\  |              |
         \                                    \\1 + x /                \1 + x /                   \1 + x /  /                                                                                                                                    \                       \1 + x /  /                                               \                       \1 + x /  /              /

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

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