Производная tan(cot(x)-2)^(3)

Вы ввели

   3            
tan (cot(x) - 2)

$$\tan^{3}{\left (\cot{\left (x \right )} - 2 \right )}$$

Подробное решение

1. Заменим $u = \tan{\left (\cot{\left (x \right )} - 2 \right )}$.

2. В силу правила, применим: $u^{3}$ получим $3 u^{2}$

3. Затем примените цепочку правил. Умножим на $\frac{d}{d x} \tan{\left (\cot{\left (x \right )} - 2 \right )}$:

   1. Есть несколько способов вычислить эту производную.

      Один из способов:

      1. Заменим $u = \cot{\left (x \right )} - 2$.

      2. $\frac{d}{d u} \tan{\left (u \right )} = \frac{1}{\cos^{2}{\left (u \right )}}$

      3. Затем примените цепочку правил. Умножим на $\frac{d}{d x}\left(\cot{\left (x \right )} - 2\right)$:

         1. дифференцируем $\cot{\left (x \right )} - 2$ почленно:

            1. Есть несколько способов вычислить эту производную.

               Один из способов:

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

            2. Производная постоянной $-2$ равна нулю.

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

         В результате последовательности правил:

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

   В результате последовательности правил:

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

4. Теперь упростим:

   $- \frac{3 \tan^{2}{\left (\cot{\left (x \right )} - 2 \right )}}{\cos^{2}{\left (x \right )} \cos^{2}{\left (\cot{\left (x \right )} - 2 \right )} \tan^{2}{\left (x \right )}}$

---

Ответ:

$- \frac{3 \tan^{2}{\left (\cot{\left (x \right )} - 2 \right )}}{\cos^{2}{\left (x \right )} \cos^{2}{\left (\cot{\left (x \right )} - 2 \right )} \tan^{2}{\left (x \right )}}$

Первая производная

     2             /       2            \ /        2   \
3*tan (cot(x) - 2)*\1 + tan (cot(x) - 2)/*\-1 - cot (x)/

$$3 \left(\tan^{2}{\left (\cot{\left (x \right )} - 2 \right )} + 1\right) \left(- \cot^{2}{\left (x \right )} - 1\right) \tan^{2}{\left (\cot{\left (x \right )} - 2 \right )}$$

Вторая производная

  /       2   \ /       2             \ /   2              /       2   \   /       2   \ /       2             \                          \                 
6*\1 + cot (x)/*\1 + tan (-2 + cot(x))/*\tan (-2 + cot(x))*\1 + cot (x)/ + \1 + cot (x)/*\1 + tan (-2 + cot(x))/ + cot(x)*tan(-2 + cot(x))/*tan(-2 + cot(x))

$$6 \left(\tan^{2}{\left (\cot{\left (x \right )} - 2 \right )} + 1\right) \left(\cot^{2}{\left (x \right )} + 1\right) \left(\left(\tan^{2}{\left (\cot{\left (x \right )} - 2 \right )} + 1\right) \left(\cot^{2}{\left (x \right )} + 1\right) + \left(\cot^{2}{\left (x \right )} + 1\right) \tan^{2}{\left (\cot{\left (x \right )} - 2 \right )} + \tan{\left (\cot{\left (x \right )} - 2 \right )} \cot{\left (x \right )}\right) \tan{\left (\cot{\left (x \right )} - 2 \right )}$$

Третья производная

                                         /             2                        2                                                    2                                                                                                             2                                                                                                            \
   /       2   \ /       2             \ |/       2   \  /       2             \       2              /       2   \     /       2   \     4                     2       2                     3              /       2   \            /       2   \     2              /       2             \     /       2   \ /       2             \                        |
-6*\1 + cot (x)/*\1 + tan (-2 + cot(x))/*\\1 + cot (x)/ *\1 + tan (-2 + cot(x))/  + tan (-2 + cot(x))*\1 + cot (x)/ + 2*\1 + cot (x)/ *tan (-2 + cot(x)) + 2*cot (x)*tan (-2 + cot(x)) + 6*tan (-2 + cot(x))*\1 + cot (x)/*cot(x) + 7*\1 + cot (x)/ *tan (-2 + cot(x))*\1 + tan (-2 + cot(x))/ + 6*\1 + cot (x)/*\1 + tan (-2 + cot(x))/*cot(x)*tan(-2 + cot(x))/

$$- 6 \left(\tan^{2}{\left (\cot{\left (x \right )} - 2 \right )} + 1\right) \left(\cot^{2}{\left (x \right )} + 1\right) \left(\left(\tan^{2}{\left (\cot{\left (x \right )} - 2 \right )} + 1\right)^{2} \left(\cot^{2}{\left (x \right )} + 1\right)^{2} + 7 \left(\tan^{2}{\left (\cot{\left (x \right )} - 2 \right )} + 1\right) \left(\cot^{2}{\left (x \right )} + 1\right)^{2} \tan^{2}{\left (\cot{\left (x \right )} - 2 \right )} + 6 \left(\tan^{2}{\left (\cot{\left (x \right )} - 2 \right )} + 1\right) \left(\cot^{2}{\left (x \right )} + 1\right) \tan{\left (\cot{\left (x \right )} - 2 \right )} \cot{\left (x \right )} + 2 \left(\cot^{2}{\left (x \right )} + 1\right)^{2} \tan^{4}{\left (\cot{\left (x \right )} - 2 \right )} + 6 \left(\cot^{2}{\left (x \right )} + 1\right) \tan^{3}{\left (\cot{\left (x \right )} - 2 \right )} \cot{\left (x \right )} + \left(\cot^{2}{\left (x \right )} + 1\right) \tan^{2}{\left (\cot{\left (x \right )} - 2 \right )} + 2 \tan^{2}{\left (\cot{\left (x \right )} - 2 \right )} \cot^{2}{\left (x \right )}\right)$$