Производная log(tanh(x/2))

Вы ввели

   /    /x\\
log|tanh|-||
   \    \2//

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

Первая производная

        2/x\
    tanh |-|
1        \2/
- - --------
2      2    
------------
      /x\   
  tanh|-|   
      \2/   

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

Вторая производная

                                 2
                  /         2/x\\ 
                  |-1 + tanh |-|| 
           2/x\   \          \2// 
-2 + 2*tanh |-| - ----------------
            \2/           2/x\    
                      tanh |-|    
                           \2/    
----------------------------------
                4                 

$$\frac{1}{4} \left(- \frac{\left(\tanh^{2}{\left (\frac{x}{2} \right )} - 1\right)^{2}}{\tanh^{2}{\left (\frac{x}{2} \right )}} + 2 \tanh^{2}{\left (\frac{x}{2} \right )} - 2\right)$$

Третья производная

                /                             2                    \
                |              /         2/x\\      /         2/x\\|
                |              |-1 + tanh |-||    2*|-1 + tanh |-|||
/         2/x\\ |        /x\   \          \2//      \          \2//|
|-1 + tanh |-||*|- 2*tanh|-| - ---------------- + -----------------|
\          \2// |        \2/           3/x\                /x\     |
                |                  tanh |-|            tanh|-|     |
                \                       \2/                \2/     /
--------------------------------------------------------------------
                                 4                                  

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