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Производная:
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Производная -((x)/(x^2+441))
Идентичные выражения
sqrt((cos(x)^ два + один)/(sin(два *x)+ один))
квадратный корень из (( косинус от (x) в квадрате плюс 1) делить на ( синус от (2 умножить на x) плюс 1))
квадратный корень из (( косинус от (x) в степени два плюс один) делить на ( синус от (два умножить на x) плюс один))
√((cos(x)^2+1)/(sin(2*x)+1))
sqrt((cos(x)2+1)/(sin(2*x)+1))
sqrtcosx2+1/sin2*x+1
sqrt((cos(x)²+1)/(sin(2*x)+1))
sqrt((cos(x) в степени 2+1)/(sin(2*x)+1))
sqrt((cos(x)^2+1)/(sin(2x)+1))
sqrt((cos(x)2+1)/(sin(2x)+1))
sqrtcosx2+1/sin2x+1
sqrtcosx^2+1/sin2x+1
sqrt((cos(x)^2+1) разделить на (sin(2*x)+1))
Похожие выражения
sqrt(cos(x)^(2)+1)/sin(2*x)+1
sqrt((cos(x)^2+1)/(sin(2*x)-1))
sqrt((cos(x)^2-1)/(sin(2*x)+1))
sqrt(cos(x)^(2)+1/(sin(2*x)+1))
sqrt((cosx^2+1)/(sin(2*x)+1))

Производная функции/ sqrt((cos(x)^2+1)/(sin(2*x)+1))

Производная sqrt((cos(x)^2+1)/(sin(2*x)+1))

Решение

Вы ввели [src]

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

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

  /     ______________\
  |    /    2         |
d |   /  cos (x) + 1  |
--|  /   ------------ |
dx\\/    sin(2*x) + 1 /

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

Преобразовать

Подробное решение

Заменим $u = \frac{\cos^{2}{\left(x \right)} + 1}{\sin{\left(2 x \right)} + 1}$ .
В силу правила, применим: $\sqrt{u}$ получим $\frac{1}{2 \sqrt{u}}$
Затем примените цепочку правил. Умножим на $\frac{d}{d x} \frac{\cos^{2}{\left(x \right)} + 1}{\sin{\left(2 x \right)} + 1}$ :
1. Применим правило производной частного:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = \cos^{2}{\left(x \right)} + 1$ и $g{\left(x \right)} = \sin{\left(2 x \right)} + 1$ .
  
  Чтобы найти $\frac{d}{d x} f{\left(x \right)}$ :
  1. дифференцируем $\cos^{2}{\left(x \right)} + 1$ почленно:
    1. Производная постоянной $1$ равна нулю.
    2. Заменим $u = \cos{\left(x \right)}$ .
    3. В силу правила, применим: $u^{2}$ получим $2 u$
    4. Затем примените цепочку правил. Умножим на $\frac{d}{d x} \cos{\left(x \right)}$ :
      1. Производная косинус есть минус синус:
        
        $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
      В результате последовательности правил:
      
      $- 2 \sin{\left(x \right)} \cos{\left(x \right)}$
    В результате: $- 2 \sin{\left(x \right)} \cos{\left(x \right)}$
  Чтобы найти $\frac{d}{d x} g{\left(x \right)}$ :
  1. дифференцируем $\sin{\left(2 x \right)} + 1$ почленно:
    1. Производная постоянной $1$ равна нулю.
    2. Заменим $u = 2 x$ .
    3. Производная синуса есть косинус:
      
      $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
    4. Затем примените цепочку правил. Умножим на $\frac{d}{d x} 2 x$ :
      1. Производная произведения константы на функцию есть произведение этой константы на производную данной функции.
        
        В силу правила, применим: $x$ получим $1$
        
        Таким образом, в результате: $2$
      В результате последовательности правил:
      
      $2 \cos{\left(2 x \right)}$
    В результате: $2 \cos{\left(2 x \right)}$
  Теперь применим правило производной деления:
  
  $\frac{- 2 \left(\sin{\left(2 x \right)} + 1\right) \sin{\left(x \right)} \cos{\left(x \right)} - 2 \left(\cos^{2}{\left(x \right)} + 1\right) \cos{\left(2 x \right)}}{\left(\sin{\left(2 x \right)} + 1\right)^{2}}$
В результате последовательности правил:

$\frac{- 2 \left(\sin{\left(2 x \right)} + 1\right) \sin{\left(x \right)} \cos{\left(x \right)} - 2 \left(\cos^{2}{\left(x \right)} + 1\right) \cos{\left(2 x \right)}}{2 \left(\sin{\left(2 x \right)} + 1\right)^{2} \sqrt{\cos^{2}{\left(x \right)} + 1} \sqrt{\frac{1}{\sin{\left(2 x \right)} + 1}}}$
Теперь упростим:

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

Ответ:

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

График

Построить график f(x)

Построить график f'(x)

Первая производная [src]

    ______________    _____________ /                  /   2       \         \               
   /      1          /    2         |  cos(x)*sin(x)   \cos (x) + 1/*cos(2*x)|               
  /  ------------ *\/  cos (x) + 1 *|- ------------- - ----------------------|*(sin(2*x) + 1)
\/   sin(2*x) + 1                   |   sin(2*x) + 1                    2    |               
                                    \                     (sin(2*x) + 1)     /               
---------------------------------------------------------------------------------------------
                                            2                                                
                                         cos (x) + 1

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

Упростить

Вторая производная [src]

                   /                                                                                                                                               /   _____________                            \                                                                                                                 \
                   |                                                                                                      /                /       2   \         \ |  /        2                                |                                                                                                                 |
                   |                      /       2   \                 2      /       2   \                              |                \1 + cos (x)/*cos(2*x)| |\/  1 + cos (x) *cos(2*x)    cos(x)*sin(x)  |                                                                                                                 |
                   |   2         2      2*\1 + cos (x)/*sin(2*x)   4*cos (2*x)*\1 + cos (x)/   4*cos(x)*cos(2*x)*sin(x)   |cos(x)*sin(x) + ----------------------|*|------------------------- + ----------------|     /                /       2   \         \                   /                /       2   \         \         |
                   |sin (x) - cos (x) + ------------------------ + ------------------------- + ------------------------   \                     1 + sin(2*x)     / |       1 + sin(2*x)            _____________|     |                \1 + cos (x)/*cos(2*x)|                   |                \1 + cos (x)/*cos(2*x)|         |
    ______________ |                          1 + sin(2*x)                            2              1 + sin(2*x)                                                  |                              /        2    |   2*|cos(x)*sin(x) + ----------------------|*cos(x)*sin(x)   2*|cos(x)*sin(x) + ----------------------|*cos(2*x)|
   /      1        |                                                    (1 + sin(2*x))                                                                             \                            \/  1 + cos (x) /     \                     1 + sin(2*x)     /                   \                     1 + sin(2*x)     /         |
  /  ------------ *|--------------------------------------------------------------------------------------------------- + --------------------------------------------------------------------------------------- - -------------------------------------------------------- - ---------------------------------------------------|
\/   1 + sin(2*x)  |                                             _____________                                                                                         2                                                                             3/2                                    _____________                         |
                   |                                            /        2                                                                                      1 + cos (x)                                                             /       2   \                                      /        2                             |
                   \                                          \/  1 + cos (x)                                                                                                                                                           \1 + cos (x)/                                    \/  1 + cos (x) *(1 + sin(2*x))          /

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

Упростить

Третья производная [src]

                   /                                                                                                                                                                                                                                                                 /                                                              _____________                 _____________                                            \   /   _____________                            \                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   /   _____________                            \                                                       /   _____________                            \                                                                                  \
                   |                                                                                                                                                                                                                        /                /       2   \         \ |       2                  2              2       2           /        2                    /        2        2                                       |   |  /        2                                | /                      /       2   \                 2      /       2   \                           \                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    /                /       2   \         \ |  /        2                                |              /                /       2   \         \ |  /        2                                |                                                                                  |
                   |    /                        2                 /       2   \                 2                     3      /       2   \                                    2                         /       2   \                  \   |                \1 + cos (x)/*cos(2*x)| |    sin (x)            cos (x)        cos (x)*sin (x)    2*\/  1 + cos (x) *sin(2*x)   3*\/  1 + cos (x) *cos (2*x)       2*cos(x)*cos(2*x)*sin(x)   |   |\/  1 + cos (x) *cos(2*x)    cos(x)*sin(x)  | |   2         2      2*\1 + cos (x)/*sin(2*x)   4*cos (2*x)*\1 + cos (x)/   4*cos(x)*cos(2*x)*sin(x)|                                                                                                                                                                                                                                   /                      /       2   \                 2      /       2   \                           \                   /                      /       2   \                 2      /       2   \                           \                                                                    |                \1 + cos (x)/*cos(2*x)| |\/  1 + cos (x) *cos(2*x)    cos(x)*sin(x)  |              |                \1 + cos (x)/*cos(2*x)| |\/  1 + cos (x) *cos(2*x)    cos(x)*sin(x)  |                                                                                  |
                   |    |                   3*cos (x)*cos(2*x)   2*\1 + cos (x)/*cos(2*x)   3*sin (x)*cos(2*x)   12*cos (2*x)*\1 + cos (x)/   6*cos(x)*sin(x)*sin(2*x)   12*cos (2*x)*cos(x)*sin(x)   12*\1 + cos (x)/*cos(2*x)*sin(2*x)|   |cos(x)*sin(x) + ----------------------|*|---------------- - ---------------- - ---------------- + --------------------------- + ---------------------------- + -------------------------------|   |------------------------- + ----------------|*|sin (x) - cos (x) + ------------------------ + ------------------------- + ------------------------|             /                /       2   \         \             /                /       2   \         \                     /                /       2   \         \               /                /       2   \         \     |   2         2      2*\1 + cos (x)/*sin(2*x)   4*cos (2*x)*\1 + cos (x)/   4*cos(x)*cos(2*x)*sin(x)|                   |   2         2      2*\1 + cos (x)/*sin(2*x)   4*cos (2*x)*\1 + cos (x)/   4*cos(x)*cos(2*x)*sin(x)|              /                /       2   \         \            2*|cos(x)*sin(x) + ----------------------|*|------------------------- + ----------------|*cos(2*x)   2*|cos(x)*sin(x) + ----------------------|*|------------------------- + ----------------|*cos(x)*sin(x)     /                /       2   \         \                       |
                   |  2*|-2*cos(x)*sin(x) - ------------------ - ------------------------ + ------------------ + -------------------------- + ------------------------ + -------------------------- + ----------------------------------|   \                     1 + sin(2*x)     / |   _____________      _____________                3/2           1 + sin(2*x)                              2             _____________               |   |       1 + sin(2*x)            _____________| |                          1 + sin(2*x)                            2              1 + sin(2*x)      |        2    |                \1 + cos (x)/*cos(2*x)|        2    |                \1 + cos (x)/*cos(2*x)|        2       2    |                \1 + cos (x)/*cos(2*x)|        2      |                \1 + cos (x)/*cos(2*x)|   3*|sin (x) - cos (x) + ------------------------ + ------------------------- + ------------------------|*cos(x)*sin(x)   3*|sin (x) - cos (x) + ------------------------ + ------------------------- + ------------------------|*cos(2*x)     |                \1 + cos (x)/*cos(2*x)|              \                     1 + sin(2*x)     / |       1 + sin(2*x)            _____________|              \                     1 + sin(2*x)     / |       1 + sin(2*x)            _____________|                   |                \1 + cos (x)/*cos(2*x)|                       |
    ______________ |    |                      1 + sin(2*x)            1 + sin(2*x)            1 + sin(2*x)                         3               1 + sin(2*x)                            2                                2          |                                            |  /        2         /        2       /       2   \                                          (1 + sin(2*x))             /        2                   |   |                              /        2    | \                                                    (1 + sin(2*x))                                 /   2*cos (x)*|cos(x)*sin(x) + ----------------------|   2*sin (x)*|cos(x)*sin(x) + ----------------------|   6*cos (x)*sin (x)*|cos(x)*sin(x) + ----------------------|   2*cos (2*x)*|cos(x)*sin(x) + ----------------------|     |                          1 + sin(2*x)                            2              1 + sin(2*x)      |                   |                          1 + sin(2*x)                            2              1 + sin(2*x)      |            4*|cos(x)*sin(x) + ----------------------|*sin(2*x)                                              |                              /        2    |                                                       |                              /        2    |                 4*|cos(x)*sin(x) + ----------------------|*cos(x)*cos(2*x)*sin(x)|
   /      1        |    \                                                                                             (1 + sin(2*x))                                          (1 + sin(2*x))                   (1 + sin(2*x))           /                                            \\/  1 + cos (x)    \/  1 + cos (x)    \1 + cos (x)/                                                                   \/  1 + cos (x) *(1 + sin(2*x))/   \                            \/  1 + cos (x) /                                                                                                                   \                     1 + sin(2*x)     /             \                     1 + sin(2*x)     /                     \                     1 + sin(2*x)     /               \                     1 + sin(2*x)     /     \                                                    (1 + sin(2*x))                                 /                   \                                                    (1 + sin(2*x))                                 /              \                     1 + sin(2*x)     /                                                       \                            \/  1 + cos (x) /                                                       \                            \/  1 + cos (x) /                   \                     1 + sin(2*x)     /                       |
  /  ------------ *|- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ - ---------------------------------------------------------------------------------------------------------------------------------------------------- - -------------------------------------------------- + -------------------------------------------------- - ---------------------------------------------------------- + ---------------------------------------------------- + --------------------------------------------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------------------- + --------------------------------------------------- + -------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------- - -----------------------------------------------------------------|
\/   1 + sin(2*x)  |                                                                                                       _____________                                                                                                                                                                                                     2                                                                                                                                                                            2                                                                                                         3/2                                                  3/2                                                      5/2                                     _____________                                                                                             3/2                                                                                                _____________                                                                        _____________                                                               /       2   \                                                                                                               2                                                                             3/2                                |
                   |                                                                                                      /        2                                                                                                                                                                                                  1 + cos (x)                                                                                                                                                                  1 + cos (x)                                                                                         /       2   \                                        /       2   \                                            /       2   \                                       /        2                   2                                                                /       2   \                                                                                                  /        2                                                                           /        2                                                                   \1 + cos (x)/*(1 + sin(2*x))                                                                                   /       2   \                                                                 /       2   \                                   |
                   \                                                                                                    \/  1 + cos (x)                                                                                                                                                                                                                                                                                                                                                                                                                                                                                \1 + cos (x)/                                        \1 + cos (x)/                                            \1 + cos (x)/                                     \/  1 + cos (x) *(1 + sin(2*x))                                                                 \1 + cos (x)/                                                                                                \/  1 + cos (x) *(1 + sin(2*x))                                                      \/  1 + cos (x) *(1 + sin(2*x))                                                                                                                                                               \1 + cos (x)/                                                                 \1 + cos (x)/   *(1 + sin(2*x))                 /

$$\left(\frac{2 \left(\sin{\left(x \right)} \cos{\left(x \right)} + \frac{\left(\cos^{2}{\left(x \right)} + 1\right) \cos{\left(2 x \right)}}{\sin{\left(2 x \right)} + 1}\right) \left(\frac{\sin{\left(x \right)} \cos{\left(x \right)}}{\sqrt{\cos^{2}{\left(x \right)} + 1}} + \frac{\sqrt{\cos^{2}{\left(x \right)} + 1} \cos{\left(2 x \right)}}{\sin{\left(2 x \right)} + 1}\right) \sin{\left(x \right)} \cos{\left(x \right)}}{\left(\cos^{2}{\left(x \right)} + 1\right)^{2}} + \frac{2 \left(\sin{\left(x \right)} \cos{\left(x \right)} + \frac{\left(\cos^{2}{\left(x \right)} + 1\right) \cos{\left(2 x \right)}}{\sin{\left(2 x \right)} + 1}\right) \left(\frac{\sin{\left(x \right)} \cos{\left(x \right)}}{\sqrt{\cos^{2}{\left(x \right)} + 1}} + \frac{\sqrt{\cos^{2}{\left(x \right)} + 1} \cos{\left(2 x \right)}}{\sin{\left(2 x \right)} + 1}\right) \cos{\left(2 x \right)}}{\left(\sin{\left(2 x \right)} + 1\right) \left(\cos^{2}{\left(x \right)} + 1\right)} - \frac{\left(\sin{\left(x \right)} \cos{\left(x \right)} + \frac{\left(\cos^{2}{\left(x \right)} + 1\right) \cos{\left(2 x \right)}}{\sin{\left(2 x \right)} + 1}\right) \left(\frac{\sin^{2}{\left(x \right)}}{\sqrt{\cos^{2}{\left(x \right)} + 1}} - \frac{\cos^{2}{\left(x \right)}}{\sqrt{\cos^{2}{\left(x \right)} + 1}} - \frac{\sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}{\left(\cos^{2}{\left(x \right)} + 1\right)^{\frac{3}{2}}} + \frac{2 \sqrt{\cos^{2}{\left(x \right)} + 1} \sin{\left(2 x \right)}}{\sin{\left(2 x \right)} + 1} + \frac{2 \sin{\left(x \right)} \cos{\left(x \right)} \cos{\left(2 x \right)}}{\left(\sin{\left(2 x \right)} + 1\right) \sqrt{\cos^{2}{\left(x \right)} + 1}} + \frac{3 \sqrt{\cos^{2}{\left(x \right)} + 1} \cos^{2}{\left(2 x \right)}}{\left(\sin{\left(2 x \right)} + 1\right)^{2}}\right)}{\cos^{2}{\left(x \right)} + 1} - \frac{\left(\frac{\sin{\left(x \right)} \cos{\left(x \right)}}{\sqrt{\cos^{2}{\left(x \right)} + 1}} + \frac{\sqrt{\cos^{2}{\left(x \right)} + 1} \cos{\left(2 x \right)}}{\sin{\left(2 x \right)} + 1}\right) \left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)} + \frac{4 \sin{\left(x \right)} \cos{\left(x \right)} \cos{\left(2 x \right)}}{\sin{\left(2 x \right)} + 1} + \frac{2 \left(\cos^{2}{\left(x \right)} + 1\right) \sin{\left(2 x \right)}}{\sin{\left(2 x \right)} + 1} + \frac{4 \left(\cos^{2}{\left(x \right)} + 1\right) \cos^{2}{\left(2 x \right)}}{\left(\sin{\left(2 x \right)} + 1\right)^{2}}\right)}{\cos^{2}{\left(x \right)} + 1} + \frac{2 \left(\sin{\left(x \right)} \cos{\left(x \right)} + \frac{\left(\cos^{2}{\left(x \right)} + 1\right) \cos{\left(2 x \right)}}{\sin{\left(2 x \right)} + 1}\right) \sin^{2}{\left(x \right)}}{\left(\cos^{2}{\left(x \right)} + 1\right)^{\frac{3}{2}}} - \frac{2 \left(\sin{\left(x \right)} \cos{\left(x \right)} + \frac{\left(\cos^{2}{\left(x \right)} + 1\right) \cos{\left(2 x \right)}}{\sin{\left(2 x \right)} + 1}\right) \cos^{2}{\left(x \right)}}{\left(\cos^{2}{\left(x \right)} + 1\right)^{\frac{3}{2}}} - \frac{6 \left(\sin{\left(x \right)} \cos{\left(x \right)} + \frac{\left(\cos^{2}{\left(x \right)} + 1\right) \cos{\left(2 x \right)}}{\sin{\left(2 x \right)} + 1}\right) \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}{\left(\cos^{2}{\left(x \right)} + 1\right)^{\frac{5}{2}}} + \frac{4 \left(\sin{\left(x \right)} \cos{\left(x \right)} + \frac{\left(\cos^{2}{\left(x \right)} + 1\right) \cos{\left(2 x \right)}}{\sin{\left(2 x \right)} + 1}\right) \sin{\left(2 x \right)}}{\left(\sin{\left(2 x \right)} + 1\right) \sqrt{\cos^{2}{\left(x \right)} + 1}} - \frac{4 \left(\sin{\left(x \right)} \cos{\left(x \right)} + \frac{\left(\cos^{2}{\left(x \right)} + 1\right) \cos{\left(2 x \right)}}{\sin{\left(2 x \right)} + 1}\right) \sin{\left(x \right)} \cos{\left(x \right)} \cos{\left(2 x \right)}}{\left(\sin{\left(2 x \right)} + 1\right) \left(\cos^{2}{\left(x \right)} + 1\right)^{\frac{3}{2}}} + \frac{2 \left(\sin{\left(x \right)} \cos{\left(x \right)} + \frac{\left(\cos^{2}{\left(x \right)} + 1\right) \cos{\left(2 x \right)}}{\sin{\left(2 x \right)} + 1}\right) \cos^{2}{\left(2 x \right)}}{\left(\sin{\left(2 x \right)} + 1\right)^{2} \sqrt{\cos^{2}{\left(x \right)} + 1}} - \frac{2 \left(- 2 \sin{\left(x \right)} \cos{\left(x \right)} + \frac{3 \sin^{2}{\left(x \right)} \cos{\left(2 x \right)}}{\sin{\left(2 x \right)} + 1} + \frac{6 \sin{\left(x \right)} \sin{\left(2 x \right)} \cos{\left(x \right)}}{\sin{\left(2 x \right)} + 1} - \frac{3 \cos^{2}{\left(x \right)} \cos{\left(2 x \right)}}{\sin{\left(2 x \right)} + 1} + \frac{12 \sin{\left(x \right)} \cos{\left(x \right)} \cos^{2}{\left(2 x \right)}}{\left(\sin{\left(2 x \right)} + 1\right)^{2}} - \frac{2 \left(\cos^{2}{\left(x \right)} + 1\right) \cos{\left(2 x \right)}}{\sin{\left(2 x \right)} + 1} + \frac{12 \left(\cos^{2}{\left(x \right)} + 1\right) \sin{\left(2 x \right)} \cos{\left(2 x \right)}}{\left(\sin{\left(2 x \right)} + 1\right)^{2}} + \frac{12 \left(\cos^{2}{\left(x \right)} + 1\right) \cos^{3}{\left(2 x \right)}}{\left(\sin{\left(2 x \right)} + 1\right)^{3}}\right)}{\sqrt{\cos^{2}{\left(x \right)} + 1}} + \frac{3 \left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)} + \frac{4 \sin{\left(x \right)} \cos{\left(x \right)} \cos{\left(2 x \right)}}{\sin{\left(2 x \right)} + 1} + \frac{2 \left(\cos^{2}{\left(x \right)} + 1\right) \sin{\left(2 x \right)}}{\sin{\left(2 x \right)} + 1} + \frac{4 \left(\cos^{2}{\left(x \right)} + 1\right) \cos^{2}{\left(2 x \right)}}{\left(\sin{\left(2 x \right)} + 1\right)^{2}}\right) \sin{\left(x \right)} \cos{\left(x \right)}}{\left(\cos^{2}{\left(x \right)} + 1\right)^{\frac{3}{2}}} + \frac{3 \left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)} + \frac{4 \sin{\left(x \right)} \cos{\left(x \right)} \cos{\left(2 x \right)}}{\sin{\left(2 x \right)} + 1} + \frac{2 \left(\cos^{2}{\left(x \right)} + 1\right) \sin{\left(2 x \right)}}{\sin{\left(2 x \right)} + 1} + \frac{4 \left(\cos^{2}{\left(x \right)} + 1\right) \cos^{2}{\left(2 x \right)}}{\left(\sin{\left(2 x \right)} + 1\right)^{2}}\right) \cos{\left(2 x \right)}}{\left(\sin{\left(2 x \right)} + 1\right) \sqrt{\cos^{2}{\left(x \right)} + 1}}\right) \sqrt{\frac{1}{\sin{\left(2 x \right)} + 1}}$$

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График

Производная sqrt((cos(x)^2+1)/(sin(2*x)+1))

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Производная sqrt((cos(x)^2+1)/(sin(2*x)+1))

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