Производная log((cot(x))^(sin(x)))

Вы ввели

   /   sin(x)   \
log\cot      (x)/

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

Подробное решение

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

2. Производная $\log{\left (u \right )}$ является $\frac{1}{u}$.

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

   1. Не могу найти шаги в поиске этой производной.

      Но производная

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

   В результате последовательности правил:

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

---

Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

                                                                                              2                         3                                                  2       
                                                                                 /       2   \             /       2   \             /       2   \            /       2   \        
                        /       2   \            /       2   \                 3*\1 + cot (x)/ *cos(x)   2*\1 + cot (x)/ *sin(x)   3*\1 + cot (x)/*sin(x)   4*\1 + cot (x)/ *sin(x)
-cos(x)*log(cot(x)) + 6*\1 + cot (x)/*cos(x) - 4*\1 + cot (x)/*cot(x)*sin(x) - ----------------------- - ----------------------- + ---------------------- + -----------------------
                                                                                          2                         3                      cot(x)                    cot(x)        
                                                                                       cot (x)                   cot (x)                                                           

$$- \frac{2 \left(\cot^{2}{\left (x \right )} + 1\right)^{3}}{\cot^{3}{\left (x \right )}} \sin{\left (x \right )} + \frac{4 \left(\cot^{2}{\left (x \right )} + 1\right)^{2}}{\cot{\left (x \right )}} \sin{\left (x \right )} - \frac{3 \left(\cot^{2}{\left (x \right )} + 1\right)^{2}}{\cot^{2}{\left (x \right )}} \cos{\left (x \right )} - 4 \left(\cot^{2}{\left (x \right )} + 1\right) \sin{\left (x \right )} \cot{\left (x \right )} + \frac{3 \sin{\left (x \right )}}{\cot{\left (x \right )}} \left(\cot^{2}{\left (x \right )} + 1\right) + 6 \left(\cot^{2}{\left (x \right )} + 1\right) \cos{\left (x \right )} - \log{\left (\cot{\left (x \right )} \right )} \cos{\left (x \right )}$$