Производная x^e^sin(x)

Вы ввели

 / sin(x)\
 \e      /
x         

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

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

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

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

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

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Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

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

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

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