Производная x^(sin(x)^3)

Вы ввели

    3   
 sin (x)
x       

$$x^{\sin^{3}{\left(x \right)}}$$

  /    3   \
d | sin (x)|
--\x       /
dx          

$$\frac{d}{d x} x^{\sin^{3}{\left(x \right)}}$$

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

   $\left(\log{\left(\sin^{3}{\left(x \right)} \right)} + 1\right) \left(\sin^{3}{\left(x \right)}\right)^{\sin^{3}{\left(x \right)}}$

---

Ответ:

$\left(\log{\left(\sin^{3}{\left(x \right)} \right)} + 1\right) \left(\sin^{3}{\left(x \right)}\right)^{\sin^{3}{\left(x \right)}}$

Первая производная

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

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

Вторая производная

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

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

Третья производная

    3    /                          3                3           3                                                         2                                                  /   2                                                           \         2          \
 sin (x) |/sin(x)                  \     6      9*sin (x)   2*sin (x)        3                   2                    9*sin (x)*cos(x)        3    /sin(x)                  \ |sin (x)        2                  2             6*cos(x)*sin(x)|   18*cos (x)*sin(x)|
x       *||------ + 3*cos(x)*log(x)| *sin (x) - --------- + --------- + 6*cos (x)*log(x) - 21*sin (x)*cos(x)*log(x) - ---------------- - 3*sin (x)*|------ + 3*cos(x)*log(x)|*|------- - 6*cos (x)*log(x) + 3*sin (x)*log(x) - ---------------| + -----------------|
         |\  x                     /                x            3                                                            2                    \  x                     / |    2                                                  x       |           x        |
         \                                                      x                                                            x                                                \   x                                                           /                    /

$$x^{\sin^{3}{\left(x \right)}} \left(\left(3 \log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}\right)^{3} \sin^{6}{\left(x \right)} - 3 \cdot \left(3 \log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}\right) \left(3 \log{\left(x \right)} \sin^{2}{\left(x \right)} - 6 \log{\left(x \right)} \cos^{2}{\left(x \right)} - \frac{6 \sin{\left(x \right)} \cos{\left(x \right)}}{x} + \frac{\sin^{2}{\left(x \right)}}{x^{2}}\right) \sin^{3}{\left(x \right)} - 21 \log{\left(x \right)} \sin^{2}{\left(x \right)} \cos{\left(x \right)} + 6 \log{\left(x \right)} \cos^{3}{\left(x \right)} - \frac{9 \sin^{3}{\left(x \right)}}{x} + \frac{18 \sin{\left(x \right)} \cos^{2}{\left(x \right)}}{x} - \frac{9 \sin^{2}{\left(x \right)} \cos{\left(x \right)}}{x^{2}} + \frac{2 \sin^{3}{\left(x \right)}}{x^{3}}\right)$$

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