Производная (x+sin(x))^x

Вы ввели

            x
(x + sin(x)) 

$$\left(x + \sin{\left (x \right )}\right)^{x}$$

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

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

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Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

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

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