Производная 2^(sqrt((sin(x)^2)*x))

Вы ввели

[LaTeX]

    ___________
   /    2      
 \/  sin (x)*x 
2              

$$2^{\sqrt{x \sin^{2}{\left (x \right )}}}$$

Подробное решение

[LaTeX]

1. Заменим $u = \sqrt{x \sin^{2}{\left (x \right )}}$.

2. $\frac{d}{d u} 2^{u} = 2^{u} \log{\left (2 \right )}$

3. Затем примените цепочку правил. Умножим на $\frac{d}{d x} \sqrt{x \sin^{2}{\left (x \right )}}$:

   1. Заменим $u = x \sin^{2}{\left (x \right )}$.

   2. В силу правила, применим: $\sqrt{u}$ получим $\frac{1}{2 \sqrt{u}}$

   3. Затем примените цепочку правил. Умножим на $\frac{d}{d x}\left(x \sin^{2}{\left (x \right )}\right)$:

      1. Применяем правило производной умножения:

         $\frac{d}{d x}\left(f{\left (x \right )} g{\left (x \right )}\right) = f{\left (x \right )} \frac{d}{d x} g{\left (x \right )} + g{\left (x \right )} \frac{d}{d x} f{\left (x \right )}$

         $f{\left (x \right )} = \sin^{2}{\left (x \right )}$; найдём $\frac{d}{d x} f{\left (x \right )}$:

         1. Заменим $u = \sin{\left (x \right )}$.

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

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

            1. Производная синуса есть косинус:

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

            В результате последовательности правил:

            $2 \sin{\left (x \right )} \cos{\left (x \right )}$

         $g{\left (x \right )} = x$; найдём $\frac{d}{d x} g{\left (x \right )}$:

         1. В силу правила, применим: $x$ получим $1$

         В результате: $2 x \sin{\left (x \right )} \cos{\left (x \right )} + \sin^{2}{\left (x \right )}$

      В результате последовательности правил:

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

   В результате последовательности правил:

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

4. Теперь упростим:

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

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Ответ:

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

Первая производная

[LaTeX]

    ___________                                                  
   /    2          ___________ /   2                     \       
 \/  sin (x)*x    /      2     |sin (x)                  |       
2              *\/  x*sin (x) *|------- + x*cos(x)*sin(x)|*log(2)
                               \   2                     /       
-----------------------------------------------------------------
                                 2                               
                            x*sin (x)                            

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

Упростить

Вторая производная

[LaTeX]

    ___________ /                                   ___________                                                ___________                                   ___________                            ___________                       \       
   /      2     |                     2            /      2     /     2           2                     \     /      2                                      /      2                               /      2                          2|       
 \/  x*sin (x)  |(2*x*cos(x) + sin(x)) *log(2)   \/  x*sin (x) *\x*cos (x) - x*sin (x) + 2*cos(x)*sin(x)/   \/  x*sin (x) *(2*x*cos(x) + sin(x))*cos(x)   \/  x*sin (x) *(2*x*cos(x) + sin(x))   \/  x*sin (x) *(2*x*cos(x) + sin(x)) |       
2              *|----------------------------- + -------------------------------------------------------- - ------------------------------------------- - ------------------------------------ + -------------------------------------|*log(2)
                |              4                                            2                                                    2                                     2*x*sin(x)                                    2                |       
                \                                                        sin (x)                                              sin (x)                                                                         4*x*sin (x)             /       
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                                                                                                                      x                                                                                                                       

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

Упростить

Третья производная

[LaTeX]

    ___________ /   ___________                            ___________                                                                                      ___________                              ___________                                                         ___________                                                  ___________                                                                               ___________                             ___________                                                                                                                                            ___________                                     ___________                                    ___________                                       ___________                                                                \       
   /      2     |  /      2                               /      2     /       2           2                       \                          2            /      2                                 /      2     /     2           2                     \              /      2     /     2           2                     \       /      2        2                                                   2                     /      2                          2     /      2                          3                           /     2           2                     \                                 3              /      2                                        /      2                          2            /      2                          3    2          /      2                           /     2           2                     \|       
 \/  x*sin (x)  |\/  x*sin (x) *(2*x*cos(x) + sin(x))   \/  x*sin (x) *\- 3*cos (x) + 3*sin (x) + 4*x*cos(x)*sin(x)/   3*(2*x*cos(x) + sin(x)) *log(2)   \/  x*sin (x) *(2*x*cos(x) + sin(x))   4*\/  x*sin (x) *\x*cos (x) - x*sin (x) + 2*cos(x)*sin(x)/*cos(x)   2*\/  x*sin (x) *\x*cos (x) - x*sin (x) + 2*cos(x)*sin(x)/   3*\/  x*sin (x) *cos (x)*(2*x*cos(x) + sin(x))   3*(2*x*cos(x) + sin(x)) *cos(x)*log(2)   3*\/  x*sin (x) *(2*x*cos(x) + sin(x))    \/  x*sin (x) *(2*x*cos(x) + sin(x))    3*(2*x*cos(x) + sin(x))*\x*cos (x) - x*sin (x) + 2*cos(x)*sin(x)/*log(2)   3*(2*x*cos(x) + sin(x)) *log(2)   2*\/  x*sin (x) *(2*x*cos(x) + sin(x))*cos(x)   3*\/  x*sin (x) *(2*x*cos(x) + sin(x)) *cos(x)   \/  x*sin (x) *(2*x*cos(x) + sin(x)) *log (2)   3*\/  x*sin (x) *(2*x*cos(x) + sin(x))*\x*cos (x) - x*sin (x) + 2*cos(x)*sin(x)/|       
2              *|------------------------------------ - ------------------------------------------------------------ - ------------------------------- + ------------------------------------ - ----------------------------------------------------------------- - ---------------------------------------------------------- + ---------------------------------------------- - -------------------------------------- - --------------------------------------- + ------------------------------------- + ------------------------------------------------------------------------ + ------------------------------- + --------------------------------------------- - ---------------------------------------------- + --------------------------------------------- + --------------------------------------------------------------------------------|*log(2)
                |               sin(x)                                               2                                               4*x                               2                                                        3                                                                2                                                     3                                         2*sin(x)                                   2    2                                   2    3                                                  2*sin(x)                                              8*x*sin(x)                                    2                                                3                                          8*x*sin(x)                                                             3                                      |       
                \                                                                 sin (x)                                                                             x *sin(x)                                              sin (x)                                                        x*sin (x)                                               sin (x)                                                                              4*x *sin (x)                             8*x *sin (x)                                                                                                                                              x*sin (x)                                      2*x*sin (x)                                                                                                       2*x*sin (x)                                   /       
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$$\frac{1}{x} 2^{\sqrt{x \sin^{2}{\left (x \right )}}} \left(\frac{\sqrt{x \sin^{2}{\left (x \right )}}}{\sin{\left (x \right )}} \left(2 x \cos{\left (x \right )} + \sin{\left (x \right )}\right) + \frac{3 \sqrt{x \sin^{2}{\left (x \right )}}}{\sin^{3}{\left (x \right )}} \left(2 x \cos{\left (x \right )} + \sin{\left (x \right )}\right) \cos^{2}{\left (x \right )} - \frac{4 \sqrt{x \sin^{2}{\left (x \right )}}}{\sin^{3}{\left (x \right )}} \left(- x \sin^{2}{\left (x \right )} + x \cos^{2}{\left (x \right )} + 2 \sin{\left (x \right )} \cos{\left (x \right )}\right) \cos{\left (x \right )} - \frac{\sqrt{x \sin^{2}{\left (x \right )}}}{\sin^{2}{\left (x \right )}} \left(4 x \sin{\left (x \right )} \cos{\left (x \right )} + 3 \sin^{2}{\left (x \right )} - 3 \cos^{2}{\left (x \right )}\right) - \frac{3 \log{\left (2 \right )} \cos{\left (x \right )}}{2 \sin{\left (x \right )}} \left(2 x \cos{\left (x \right )} + \sin{\left (x \right )}\right)^{2} + \frac{3 \log{\left (2 \right )}}{2 \sin{\left (x \right )}} \left(2 x \cos{\left (x \right )} + \sin{\left (x \right )}\right) \left(- x \sin^{2}{\left (x \right )} + x \cos^{2}{\left (x \right )} + 2 \sin{\left (x \right )} \cos{\left (x \right )}\right) + \frac{\sqrt{x \sin^{2}{\left (x \right )}} \log^{2}{\left (2 \right )}}{8 x \sin{\left (x \right )}} \left(2 x \cos{\left (x \right )} + \sin{\left (x \right )}\right)^{3} - \frac{3 \sqrt{x \sin^{2}{\left (x \right )}} \cos{\left (x \right )}}{2 x \sin^{3}{\left (x \right )}} \left(2 x \cos{\left (x \right )} + \sin{\left (x \right )}\right)^{2} + \frac{3 \sqrt{x \sin^{2}{\left (x \right )}}}{2 x \sin^{3}{\left (x \right )}} \left(2 x \cos{\left (x \right )} + \sin{\left (x \right )}\right) \left(- x \sin^{2}{\left (x \right )} + x \cos^{2}{\left (x \right )} + 2 \sin{\left (x \right )} \cos{\left (x \right )}\right) + \frac{2 \sqrt{x \sin^{2}{\left (x \right )}} \cos{\left (x \right )}}{x \sin^{2}{\left (x \right )}} \left(2 x \cos{\left (x \right )} + \sin{\left (x \right )}\right) - \frac{2 \sqrt{x \sin^{2}{\left (x \right )}}}{x \sin^{2}{\left (x \right )}} \left(- x \sin^{2}{\left (x \right )} + x \cos^{2}{\left (x \right )} + 2 \sin{\left (x \right )} \cos{\left (x \right )}\right) + \frac{3 \left(2 x \cos{\left (x \right )} + \sin{\left (x \right )}\right)^{3}}{8 x \sin{\left (x \right )}} \log{\left (2 \right )} - \frac{3}{4 x} \left(2 x \cos{\left (x \right )} + \sin{\left (x \right )}\right)^{2} \log{\left (2 \right )} + \frac{\sqrt{x \sin^{2}{\left (x \right )}}}{8 x^{2} \sin^{3}{\left (x \right )}} \left(2 x \cos{\left (x \right )} + \sin{\left (x \right )}\right)^{3} - \frac{3 \sqrt{x \sin^{2}{\left (x \right )}}}{4 x^{2} \sin^{2}{\left (x \right )}} \left(2 x \cos{\left (x \right )} + \sin{\left (x \right )}\right)^{2} + \frac{\sqrt{x \sin^{2}{\left (x \right )}}}{x^{2} \sin{\left (x \right )}} \left(2 x \cos{\left (x \right )} + \sin{\left (x \right )}\right)\right) \log{\left (2 \right )}$$

Упростить