Производная cot(x)/tan(x)

Вы ввели

cot(x)
------
tan(x)

$$\frac{\cot{\left (x \right )}}{\tan{\left (x \right )}}$$

Подробное решение

1. Применим правило производной частного:

   $\frac{d}{d x}\left(\frac{f{\left (x \right )}}{g{\left (x \right )}}\right) = \frac{1}{g^{2}{\left (x \right )}} \left(- f{\left (x \right )} \frac{d}{d x} g{\left (x \right )} + g{\left (x \right )} \frac{d}{d x} f{\left (x \right )}\right)$

   $f{\left (x \right )} = \cot{\left (x \right )}$ и $g{\left (x \right )} = \tan{\left (x \right )}$.

   Чтобы найти $\frac{d}{d x} f{\left (x \right )}$:

   1. Есть несколько способов вычислить эту производную.

      Один из способов:

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

   Чтобы найти $\frac{d}{d x} g{\left (x \right )}$:

   1. Есть несколько способов вычислить эту производную.

      Один из способов:

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

   Теперь применим правило производной деления:

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

2. Теперь упростим:

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

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Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

  /               2                                           2                                3                                                                                   2                                              \
  |  /       2   \                               /       2   \  /       2   \     /       2   \                2    /       2   \     /       2   \ /       2   \     /       2   \             /       2   \ /       2   \       |
  |  \1 + cot (x)/      /       2   \          3*\1 + tan (x)/ *\1 + cot (x)/   3*\1 + tan (x)/ *cot(x)   2*cot (x)*\1 + cot (x)/   3*\1 + cot (x)/*\1 + tan (x)/   5*\1 + tan (x)/ *cot(x)   3*\1 + cot (x)/*\1 + tan (x)/*cot(x)|
2*|- -------------- - 2*\1 + tan (x)/*cot(x) - ------------------------------ - ----------------------- - ----------------------- + ----------------------------- + ----------------------- - ------------------------------------|
  |      tan(x)                                              3                             4                       tan(x)                       tan(x)                         2                               2                  |
  \                                                       tan (x)                       tan (x)                                                                             tan (x)                         tan (x)               /

$$2 \left(- \frac{3 \left(\tan^{2}{\left (x \right )} + 1\right)^{3}}{\tan^{4}{\left (x \right )}} \cot{\left (x \right )} - \frac{3 \left(\tan^{2}{\left (x \right )} + 1\right)^{2}}{\tan^{3}{\left (x \right )}} \left(\cot^{2}{\left (x \right )} + 1\right) + \frac{5 \left(\tan^{2}{\left (x \right )} + 1\right)^{2}}{\tan^{2}{\left (x \right )}} \cot{\left (x \right )} + \frac{3}{\tan{\left (x \right )}} \left(\tan^{2}{\left (x \right )} + 1\right) \left(\cot^{2}{\left (x \right )} + 1\right) - \frac{3 \cot{\left (x \right )}}{\tan^{2}{\left (x \right )}} \left(\tan^{2}{\left (x \right )} + 1\right) \left(\cot^{2}{\left (x \right )} + 1\right) - 2 \left(\tan^{2}{\left (x \right )} + 1\right) \cot{\left (x \right )} - \frac{\left(\cot^{2}{\left (x \right )} + 1\right)^{2}}{\tan{\left (x \right )}} - \frac{2 \cot^{2}{\left (x \right )}}{\tan{\left (x \right )}} \left(\cot^{2}{\left (x \right )} + 1\right)\right)$$