Производная x^sin(sin(sin(x)))

Вы ввели

 sin(sin(sin(x)))
x                

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

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

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

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

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

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Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

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

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

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