Производная sin(cot(x))^(6)

Вы ввели

   6        
sin (cot(x))

$$\sin^{6}{\left (\cot{\left (x \right )} \right )}$$

Подробное решение

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

2. В силу правила, применим: $u^{6}$ получим $6 u^{5}$

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

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

   2. Производная синуса есть косинус:

      $\frac{d}{d u} \sin{\left (u \right )} = \cos{\left (u \right )}$

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

      1. Есть несколько способов вычислить эту производную.

         Один из способов:

         1. $\frac{d}{d x} \cot{\left (x \right )} = - \frac{1}{\sin^{2}{\left (x \right )}}$

      В результате последовательности правил:

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

   В результате последовательности правил:

   $- \frac{6 \sin^{5}{\left (\cot{\left (x \right )} \right )} \cos{\left (\cot{\left (x \right )} \right )}}{\cos^{2}{\left (x \right )} \tan^{2}{\left (x \right )}} \left(\sin^{2}{\left (x \right )} + \cos^{2}{\left (x \right )}\right)$

4. Теперь упростим:

   $- \frac{6 \sin^{5}{\left (\frac{1}{\tan{\left (x \right )}} \right )}}{\sin^{2}{\left (x \right )}} \cos{\left (\frac{1}{\tan{\left (x \right )}} \right )}$

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Ответ:

$- \frac{6 \sin^{5}{\left (\frac{1}{\tan{\left (x \right )}} \right )}}{\sin^{2}{\left (x \right )}} \cos{\left (\frac{1}{\tan{\left (x \right )}} \right )}$

Первая производная

     5         /        2   \            
6*sin (cot(x))*\-1 - cot (x)/*cos(cot(x))

$$6 \left(- \cot^{2}{\left (x \right )} - 1\right) \sin^{5}{\left (\cot{\left (x \right )} \right )} \cos{\left (\cot{\left (x \right )} \right )}$$

Вторая производная

     4         /       2   \ /     2         /       2   \        2         /       2   \                                   \
6*sin (cot(x))*\1 + cot (x)/*\- sin (cot(x))*\1 + cot (x)/ + 5*cos (cot(x))*\1 + cot (x)/ + 2*cos(cot(x))*cot(x)*sin(cot(x))/

$$6 \left(\cot^{2}{\left (x \right )} + 1\right) \left(- \left(\cot^{2}{\left (x \right )} + 1\right) \sin^{2}{\left (\cot{\left (x \right )} \right )} + 5 \left(\cot^{2}{\left (x \right )} + 1\right) \cos^{2}{\left (\cot{\left (x \right )} \right )} + 2 \sin{\left (\cot{\left (x \right )} \right )} \cos{\left (\cot{\left (x \right )} \right )} \cot{\left (x \right )}\right) \sin^{4}{\left (\cot{\left (x \right )} \right )}$$

Третья производная

                              /                  2                                                                                                                                                   2                                                                            \
      3         /       2   \ |     /       2   \     3              2         /       2   \                    2       2                            3         /       2   \            /       2   \     2                             2         /       2   \                   |
12*sin (cot(x))*\1 + cot (x)/*\- 10*\1 + cot (x)/ *cos (cot(x)) - sin (cot(x))*\1 + cot (x)/*cos(cot(x)) - 2*cot (x)*sin (cot(x))*cos(cot(x)) + 3*sin (cot(x))*\1 + cot (x)/*cot(x) + 8*\1 + cot (x)/ *sin (cot(x))*cos(cot(x)) - 15*cos (cot(x))*\1 + cot (x)/*cot(x)*sin(cot(x))/

$$12 \left(\cot^{2}{\left (x \right )} + 1\right) \left(8 \left(\cot^{2}{\left (x \right )} + 1\right)^{2} \sin^{2}{\left (\cot{\left (x \right )} \right )} \cos{\left (\cot{\left (x \right )} \right )} - 10 \left(\cot^{2}{\left (x \right )} + 1\right)^{2} \cos^{3}{\left (\cot{\left (x \right )} \right )} + 3 \left(\cot^{2}{\left (x \right )} + 1\right) \sin^{3}{\left (\cot{\left (x \right )} \right )} \cot{\left (x \right )} - \left(\cot^{2}{\left (x \right )} + 1\right) \sin^{2}{\left (\cot{\left (x \right )} \right )} \cos{\left (\cot{\left (x \right )} \right )} - 15 \left(\cot^{2}{\left (x \right )} + 1\right) \sin{\left (\cot{\left (x \right )} \right )} \cos^{2}{\left (\cot{\left (x \right )} \right )} \cot{\left (x \right )} - 2 \sin^{2}{\left (\cot{\left (x \right )} \right )} \cos{\left (\cot{\left (x \right )} \right )} \cot^{2}{\left (x \right )}\right) \sin^{3}{\left (\cot{\left (x \right )} \right )}$$