Производная sin(cos(cot(x)))^(2)

Вы ввели

   2             
sin (cos(cot(x)))

$$\sin^{2}{\left (\cos{\left (\cot{\left (x \right )} \right )} \right )}$$

Подробное решение

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

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

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

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

   2. Производная синуса есть косинус:

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

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

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

      2. Производная косинус есть минус синус:

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

      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:

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

      В результате последовательности правил:

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

   В результате последовательности правил:

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

4. Теперь упростим:

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

---

Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

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

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