Производная 3^x^cos(x)

Вы ввели

 / cos(x)\
 \x      /
3         

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

Подробное решение

1. Заменим $u = x^{\cos{\left (x \right )}}$.

2. $\frac{d}{d u} 3^{u} = 3^{u} \log{\left (3 \right )}$

3. Затем примените цепочку правил. Умножим на $\frac{d}{d x} x^{\cos{\left (x \right )}}$:

   1. Не могу найти шаги в поиске этой производной.

      Но производная

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

   В результате последовательности правил:

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

---

Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

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

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