Производная (tan(x)+cot(x))^7

Вы ввели

                 7
(tan(x) + cot(x)) 

$$\left(\tan{\left (x \right )} + \cot{\left (x \right )}\right)^{7}$$

Подробное решение

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

2. В силу правила, применим: $u^{7}$ получим $7 u^{6}$

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

   1. дифференцируем $\tan{\left (x \right )} + \cot{\left (x \right )}$ почленно:

      1. Есть несколько способов вычислить эту производную.

         Один из способов:

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

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

   В результате последовательности правил:

   $7 \left(\frac{1}{\cos^{2}{\left (x \right )}} \left(\sin^{2}{\left (x \right )} + \cos^{2}{\left (x \right )}\right) - \frac{\sin^{2}{\left (x \right )} + \cos^{2}{\left (x \right )}}{\cos^{2}{\left (x \right )} \tan^{2}{\left (x \right )}}\right) \left(\tan{\left (x \right )} + \cot{\left (x \right )}\right)^{6}$

4. Теперь упростим:

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

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Ответ:

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

Первая производная

                 6 /       2           2   \
(tan(x) + cot(x)) *\- 7*cot (x) + 7*tan (x)/

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

Вторая производная

                      /                     2                                                                  \
                    5 |  /   2         2   \    //       2   \          /       2   \       \                  |
14*(cot(x) + tan(x)) *\3*\tan (x) - cot (x)/  + \\1 + cot (x)/*cot(x) + \1 + tan (x)/*tan(x)/*(cot(x) + tan(x))/

$$14 \left(\left(\left(\tan^{2}{\left (x \right )} + 1\right) \tan{\left (x \right )} + \left(\cot^{2}{\left (x \right )} + 1\right) \cot{\left (x \right )}\right) \left(\tan{\left (x \right )} + \cot{\left (x \right )}\right) + 3 \left(\tan^{2}{\left (x \right )} - \cot^{2}{\left (x \right )}\right)^{2}\right) \left(\tan{\left (x \right )} + \cot{\left (x \right )}\right)^{5}$$

Третья производная

                      /                      3                      /             2                2                                                    \                                                                                         \
                    4 |   /   2         2   \                     2 |/       2   \    /       2   \         2    /       2   \        2    /       2   \|      /   2         2   \ //       2   \          /       2   \       \                  |
14*(cot(x) + tan(x)) *\15*\tan (x) - cot (x)/  + (cot(x) + tan(x)) *\\1 + tan (x)/  - \1 + cot (x)/  - 2*cot (x)*\1 + cot (x)/ + 2*tan (x)*\1 + tan (x)// + 18*\tan (x) - cot (x)/*\\1 + cot (x)/*cot(x) + \1 + tan (x)/*tan(x)/*(cot(x) + tan(x))/

$$14 \left(\tan{\left (x \right )} + \cot{\left (x \right )}\right)^{4} \left(18 \left(\left(\tan^{2}{\left (x \right )} + 1\right) \tan{\left (x \right )} + \left(\cot^{2}{\left (x \right )} + 1\right) \cot{\left (x \right )}\right) \left(\tan{\left (x \right )} + \cot{\left (x \right )}\right) \left(\tan^{2}{\left (x \right )} - \cot^{2}{\left (x \right )}\right) + \left(\tan{\left (x \right )} + \cot{\left (x \right )}\right)^{2} \left(\left(\tan^{2}{\left (x \right )} + 1\right)^{2} + 2 \left(\tan^{2}{\left (x \right )} + 1\right) \tan^{2}{\left (x \right )} - \left(\cot^{2}{\left (x \right )} + 1\right)^{2} - 2 \left(\cot^{2}{\left (x \right )} + 1\right) \cot^{2}{\left (x \right )}\right) + 15 \left(\tan^{2}{\left (x \right )} - \cot^{2}{\left (x \right )}\right)^{3}\right)$$