Производная cot(x)/sin(x)

Вы ввели

cot(x)
------
sin(x)

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

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 $g{\left (x \right )} = \sin{\left (x \right )}$.

   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. Производная синуса есть косинус:

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

   Теперь применим правило производной деления:

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

2. Теперь упростим:

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

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Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

 /                   2                                                                2    /       2   \        3               /       2   \              \ 
 |      /       2   \         2           2    /       2   \   5*cos(x)*cot(x)   6*cos (x)*\1 + cot (x)/   6*cos (x)*cot(x)   6*\1 + cot (x)/*cos(x)*cot(x)| 
-|3 + 2*\1 + cot (x)/  + 3*cot (x) + 4*cot (x)*\1 + cot (x)/ + --------------- + ----------------------- + ---------------- + -----------------------------| 
 |                                                                  sin(x)                  2                     3                       sin(x)           | 
 \                                                                                       sin (x)               sin (x)                                     / 
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                                                                            sin(x)                                                                           

$$- \frac{1}{\sin{\left (x \right )}} \left(2 \left(\cot^{2}{\left (x \right )} + 1\right)^{2} + 4 \left(\cot^{2}{\left (x \right )} + 1\right) \cot^{2}{\left (x \right )} + \frac{6 \cos{\left (x \right )}}{\sin{\left (x \right )}} \left(\cot^{2}{\left (x \right )} + 1\right) \cot{\left (x \right )} + \frac{6 \cos^{2}{\left (x \right )}}{\sin^{2}{\left (x \right )}} \left(\cot^{2}{\left (x \right )} + 1\right) + 3 \cot^{2}{\left (x \right )} + 3 + \frac{5 \cos{\left (x \right )}}{\sin{\left (x \right )}} \cot{\left (x \right )} + \frac{6 \cos^{3}{\left (x \right )}}{\sin^{3}{\left (x \right )}} \cot{\left (x \right )}\right)$$