Производная log(2,sin(x/2))

Вы ввели

   /      /x\\
log|2, sin|-||
   \      \2//

$$\log{\left (2,\sin{\left (\frac{x}{2} \right )} \right )}$$

Первая производная

/  d  /  log(2) \\|               /x\
|-----|---------|||        /x\*cos|-|
\dxi_2\log(xi_2)//|xi_2=sin|-|    \2/
                           \2/       
-------------------------------------
                  2                  

$$\frac{1}{2} \cos{\left (\frac{x}{2} \right )} \left. \frac{d}{d \xi_{2}}\left(\frac{\log{\left (2 \right )}}{\log{\left (\xi_{2} \right )}}\right) \right|_{\substack{ \xi_{2}=\sin{\left (\frac{x}{2} \right )} }}$$

Вторая производная

                                             2/x\ /         2     \       
                                          cos |-|*|1 + -----------|*log(2)
                                              \2/ |       /   /x\\|       
                                                  |    log|sin|-|||       
  /  d  /  log(2) \\|               /x\           \       \   \2///       
- |-----|---------|||        /x\*sin|-| + --------------------------------
  \dxi_2\log(xi_2)//|xi_2=sin|-|    \2/            2/   /x\\    2/x\      
                             \2/                log |sin|-||*sin |-|      
                                                    \   \2//     \2/      
--------------------------------------------------------------------------
                                    4                                     

$$\frac{1}{4} \left(\frac{\left(1 + \frac{2}{\log{\left (\sin{\left (\frac{x}{2} \right )} \right )}}\right) \log{\left (2 \right )} \cos^{2}{\left (\frac{x}{2} \right )}}{\log^{2}{\left (\sin{\left (\frac{x}{2} \right )} \right )} \sin^{2}{\left (\frac{x}{2} \right )}} - \sin{\left (\frac{x}{2} \right )} \left. \frac{d}{d \xi_{2}}\left(\frac{\log{\left (2 \right )}}{\log{\left (\xi_{2} \right )}}\right) \right|_{\substack{ \xi_{2}=\sin{\left (\frac{x}{2} \right )} }}\right)$$

Третья производная

 /                         /         2     \               2/x\ /         2     \               2/x\ /         2     \                                        \        
 |                       3*|1 + -----------|*log(2)   2*cos |-|*|1 + -----------|*log(2)   2*cos |-|*|1 + -----------|*log(2)                                 |        
 |       2/x\              |       /   /x\\|                \2/ |       /   /x\\|                \2/ |       /   /x\\|                                        |        
 |  2*cos |-|*log(2)       |    log|sin|-|||                    |    log|sin|-|||                    |    log|sin|-|||                                        |        
 |        \2/              \       \   \2///                    \       \   \2///                    \       \   \2///          /  d  /  log(2) \\|           |    /x\ 
-|-------------------- + -------------------------- + ---------------------------------- + ---------------------------------- + |-----|---------|||        /x\|*cos|-| 
 |   4/   /x\\    3/x\         2/   /x\\    /x\                 3/   /x\\    3/x\                    2/   /x\\    3/x\          \dxi_2\log(xi_2)//|xi_2=sin|-||    \2/ 
 |log |sin|-||*sin |-|      log |sin|-||*sin|-|              log |sin|-||*sin |-|                 log |sin|-||*sin |-|                                     \2/|        
 \    \   \2//     \2/          \   \2//    \2/                  \   \2//     \2/                     \   \2//     \2/                                        /        
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                   8                                                                                   

$$- \frac{1}{8} \left(\frac{3 \left(1 + \frac{2}{\log{\left (\sin{\left (\frac{x}{2} \right )} \right )}}\right) \log{\left (2 \right )}}{\log^{2}{\left (\sin{\left (\frac{x}{2} \right )} \right )} \sin{\left (\frac{x}{2} \right )}} + \frac{2 \left(1 + \frac{2}{\log{\left (\sin{\left (\frac{x}{2} \right )} \right )}}\right) \log{\left (2 \right )} \cos^{2}{\left (\frac{x}{2} \right )}}{\log^{2}{\left (\sin{\left (\frac{x}{2} \right )} \right )} \sin^{3}{\left (\frac{x}{2} \right )}} + \frac{2 \left(1 + \frac{2}{\log{\left (\sin{\left (\frac{x}{2} \right )} \right )}}\right) \log{\left (2 \right )} \cos^{2}{\left (\frac{x}{2} \right )}}{\log^{3}{\left (\sin{\left (\frac{x}{2} \right )} \right )} \sin^{3}{\left (\frac{x}{2} \right )}} + \left. \frac{d}{d \xi_{2}}\left(\frac{\log{\left (2 \right )}}{\log{\left (\xi_{2} \right )}}\right) \right|_{\substack{ \xi_{2}=\sin{\left (\frac{x}{2} \right )} }} + \frac{2 \log{\left (2 \right )} \cos^{2}{\left (\frac{x}{2} \right )}}{\log^{4}{\left (\sin{\left (\frac{x}{2} \right )} \right )} \sin^{3}{\left (\frac{x}{2} \right )}}\right) \cos{\left (\frac{x}{2} \right )}$$