Производная tan(cot(x))^(2)

Вы ввели

   2        
tan (cot(x))

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

Подробное решение

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

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

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

   1. Есть несколько способов вычислить эту производную.

      Один из способов:

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

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

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

         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:

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

   В результате последовательности правил:

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

4. Теперь упростим:

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

---

Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

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

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