Производная cot(x)/(1+cot(x))

Вы ввели

  cot(x)  
----------
1 + cot(x)

$$\frac{\cot{\left (x \right )}}{\cot{\left (x \right )} + 1}$$

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 $g{\left (x \right )} = \cot{\left (x \right )} + 1$.

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

      1. Производная постоянной $1$ равна нулю.

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

   Теперь применим правило производной деления:

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

2. Теперь упростим:

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

---

Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

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

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