Производная (sin(x)/x)^x

Вы ввели

        x
/sin(x)\ 
|------| 
\  x   / 

$$\left(\frac{1}{x} \sin{\left (x \right )}\right)^{x}$$

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

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

---

Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

          /                                                                                                                                                                                                                                                  /  2*sin(x)   2*cos(x)         \                                                                                                                                            \
          |                                                                                                                                           /  sin(x)         \                 /  sin(x)         \                 2    /  sin(x)         \   2*x*|- -------- + -------- + sin(x)|*cos(x)     /  /  sin(x)         \              \ /                                                              /  sin(x)         \       \|
          |                                                                                                                                         3*|- ------ + cos(x)|               5*|- ------ + cos(x)|*cos(x)   2*x*cos (x)*|- ------ + cos(x)|       |      2         x             |            |x*|- ------ + cos(x)|              | |                                                            x*|- ------ + cos(x)|*cos(x)||
          |                                     3                 /  sin(x)         \     /  6*cos(x)   3*sin(x)   6*sin(x)         \   10*cos(x)     \    x            /   10*sin(x)     \    x            /                      \    x            /       \     x                        /            |  \    x            /      /sin(x)\| |              /  2*sin(x)   2*cos(x)         \   3*sin(x)     \    x            /       ||
          |/  /  sin(x)         \              \    -5*sin(x) + x*|- ------ + cos(x)| - x*|- -------- - -------- + -------- + cos(x)| - --------- + --------------------- + --------- - ---------------------------- + ------------------------------- + -------------------------------------------   3*|--------------------- + log|------||*|-3*cos(x) + x*|- -------- + -------- + sin(x)| + -------- + ----------------------------||
        x ||x*|- ------ + cos(x)|              |                  \    x            /     |      2         x           3            |       x                 x                  2                 sin(x)                             2                                     sin(x)                       \        sin(x)             \  x   // |              |      2         x             |      x                  sin(x)           ||
/sin(x)\  ||  \    x            /      /sin(x)\|                                          \     x                     x             /                                           x                                                  sin (x)                                                                                                     \              \     x                        /                                          /|
|------| *||--------------------- + log|------||  + ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ - ----------------------------------------------------------------------------------------------------------------------------------|
\  x   /  \\        sin(x)             \  x   //                                                                                                                         sin(x)                                                                                                                                                                                      sin(x)                                                              /

$$\left(\frac{1}{x} \sin{\left (x \right )}\right)^{x} \left(\left(\frac{x}{\sin{\left (x \right )}} \left(\cos{\left (x \right )} - \frac{1}{x} \sin{\left (x \right )}\right) + \log{\left (\frac{1}{x} \sin{\left (x \right )} \right )}\right)^{3} - \frac{3}{\sin{\left (x \right )}} \left(\frac{x}{\sin{\left (x \right )}} \left(\cos{\left (x \right )} - \frac{1}{x} \sin{\left (x \right )}\right) + \log{\left (\frac{1}{x} \sin{\left (x \right )} \right )}\right) \left(\frac{x \cos{\left (x \right )}}{\sin{\left (x \right )}} \left(\cos{\left (x \right )} - \frac{1}{x} \sin{\left (x \right )}\right) + x \left(\sin{\left (x \right )} + \frac{2}{x} \cos{\left (x \right )} - \frac{2}{x^{2}} \sin{\left (x \right )}\right) - 3 \cos{\left (x \right )} + \frac{3}{x} \sin{\left (x \right )}\right) + \frac{1}{\sin{\left (x \right )}} \left(x \left(\cos{\left (x \right )} - \frac{1}{x} \sin{\left (x \right )}\right) + \frac{2 x \cos^{2}{\left (x \right )}}{\sin^{2}{\left (x \right )}} \left(\cos{\left (x \right )} - \frac{1}{x} \sin{\left (x \right )}\right) + \frac{2 x \cos{\left (x \right )}}{\sin{\left (x \right )}} \left(\sin{\left (x \right )} + \frac{2}{x} \cos{\left (x \right )} - \frac{2}{x^{2}} \sin{\left (x \right )}\right) - x \left(\cos{\left (x \right )} - \frac{3}{x} \sin{\left (x \right )} - \frac{6}{x^{2}} \cos{\left (x \right )} + \frac{6}{x^{3}} \sin{\left (x \right )}\right) - \frac{5 \cos{\left (x \right )}}{\sin{\left (x \right )}} \left(\cos{\left (x \right )} - \frac{1}{x} \sin{\left (x \right )}\right) - 5 \sin{\left (x \right )} + \frac{1}{x} \left(3 \cos{\left (x \right )} - \frac{3}{x} \sin{\left (x \right )}\right) - \frac{10}{x} \cos{\left (x \right )} + \frac{10}{x^{2}} \sin{\left (x \right )}\right)\right)$$