Производная sin(x)/(|x|)

Вы ввели

sin(x)
------
 |x|  

$$\frac{1}{\left|{x}\right|} \sin{\left (x \right )}$$

Первая производная

         /d                 d              \       
         |--(im(x))*im(x) + --(re(x))*re(x)|*sin(x)
cos(x)   \dx                dx             /       
------ - ------------------------------------------
 |x|                           3                   
                            |x|                    

$$- \frac{1}{\left|{x}\right|^{3}} \left(\Re{x} \frac{d}{d x} \Re{x} + \Im{x} \frac{d}{d x} \Im{x}\right) \sin{\left (x \right )} + \frac{1}{\left|{x}\right|} \cos{\left (x \right )}$$

Вторая производная

          /           2              2     2                  2             \                                                                                                      
          |/d        \    /d        \     d                  d              |                                                                                              2       
          ||--(im(x))|  + |--(re(x))|  + ---(im(x))*im(x) + ---(re(x))*re(x)|*sin(x)     /d                 d              \            /d                 d              \        
          |\dx       /    \dx       /      2                  2             |          2*|--(im(x))*im(x) + --(re(x))*re(x)|*cos(x)   3*|--(im(x))*im(x) + --(re(x))*re(x)| *sin(x)
          \                              dx                 dx              /            \dx                dx             /            \dx                dx             /        
-sin(x) - -------------------------------------------------------------------------- - -------------------------------------------- + ---------------------------------------------
                                                2                                                             2                                               4                    
                                             |x|                                                           |x|                                             |x|                     
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                        |x|                                                                                        

$$\frac{1}{\left|{x}\right|} \left(\frac{3}{\left|{x}\right|^{4}} \left(\Re{x} \frac{d}{d x} \Re{x} + \Im{x} \frac{d}{d x} \Im{x}\right)^{2} \sin{\left (x \right )} - \frac{2}{\left|{x}\right|^{2}} \left(\Re{x} \frac{d}{d x} \Re{x} + \Im{x} \frac{d}{d x} \Im{x}\right) \cos{\left (x \right )} - \frac{1}{\left|{x}\right|^{2}} \left(\Re{x} \frac{d^{2}}{d x^{2}} \Re{x} + \Im{x} \frac{d^{2}}{d x^{2}} \Im{x} + \frac{d}{d x} \Re{x}^{2} + \frac{d}{d x} \Im{x}^{2}\right) \sin{\left (x \right )} - \sin{\left (x \right )}\right)$$

Третья производная

          /  3                  3                              2                        2       \                                                             /           2              2     2                  2             \                                                                                                                                               /           2              2     2                  2             \       
          | d                  d                   d          d             d          d        |                                                3            |/d        \    /d        \     d                  d              |                                                                                              2            /d                 d              \ |/d        \    /d        \     d                  d              |       
          |---(im(x))*im(x) + ---(re(x))*re(x) + 3*--(im(x))*---(im(x)) + 3*--(re(x))*---(re(x))|*sin(x)      /d                 d              \           3*||--(im(x))|  + |--(re(x))|  + ---(im(x))*im(x) + ---(re(x))*re(x)|*cos(x)     /d                 d              \            /d                 d              \           9*|--(im(x))*im(x) + --(re(x))*re(x)|*||--(im(x))|  + |--(re(x))|  + ---(im(x))*im(x) + ---(re(x))*re(x)|*sin(x)
          |  3                  3                  dx          2            dx          2       |          15*|--(im(x))*im(x) + --(re(x))*re(x)| *sin(x)     |\dx       /    \dx       /      2                  2             |          3*|--(im(x))*im(x) + --(re(x))*re(x)|*sin(x)   9*|--(im(x))*im(x) + --(re(x))*re(x)| *cos(x)     \dx                dx             / |\dx       /    \dx       /      2                  2             |       
          \dx                 dx                             dx                       dx        /             \dx                dx             /             \                              dx                 dx              /            \dx                dx             /            \dx                dx             /                                                 \                              dx                 dx              /       
-cos(x) - ---------------------------------------------------------------------------------------------- - ---------------------------------------------- - ---------------------------------------------------------------------------- + -------------------------------------------- + --------------------------------------------- + ----------------------------------------------------------------------------------------------------------------
                                                          2                                                                        6                                                               2                                                              2                                               4                                                                                4                                                      
                                                       |x|                                                                      |x|                                                             |x|                                                            |x|                                             |x|                                                                              |x|                                                       
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                                                                                                           |x|                                                                                                                                                                                                                            

$$\frac{1}{\left|{x}\right|} \left(- \frac{15}{\left|{x}\right|^{6}} \left(\Re{x} \frac{d}{d x} \Re{x} + \Im{x} \frac{d}{d x} \Im{x}\right)^{3} \sin{\left (x \right )} + \frac{9}{\left|{x}\right|^{4}} \left(\Re{x} \frac{d}{d x} \Re{x} + \Im{x} \frac{d}{d x} \Im{x}\right)^{2} \cos{\left (x \right )} + \frac{9}{\left|{x}\right|^{4}} \left(\Re{x} \frac{d}{d x} \Re{x} + \Im{x} \frac{d}{d x} \Im{x}\right) \left(\Re{x} \frac{d^{2}}{d x^{2}} \Re{x} + \Im{x} \frac{d^{2}}{d x^{2}} \Im{x} + \frac{d}{d x} \Re{x}^{2} + \frac{d}{d x} \Im{x}^{2}\right) \sin{\left (x \right )} + \frac{3}{\left|{x}\right|^{2}} \left(\Re{x} \frac{d}{d x} \Re{x} + \Im{x} \frac{d}{d x} \Im{x}\right) \sin{\left (x \right )} - \frac{3}{\left|{x}\right|^{2}} \left(\Re{x} \frac{d^{2}}{d x^{2}} \Re{x} + \Im{x} \frac{d^{2}}{d x^{2}} \Im{x} + \frac{d}{d x} \Re{x}^{2} + \frac{d}{d x} \Im{x}^{2}\right) \cos{\left (x \right )} - \frac{1}{\left|{x}\right|^{2}} \left(\Re{x} \frac{d^{3}}{d x^{3}} \Re{x} + \Im{x} \frac{d^{3}}{d x^{3}} \Im{x} + 3 \frac{d}{d x} \Re{x} \frac{d^{2}}{d x^{2}} \Re{x} + 3 \frac{d}{d x} \Im{x} \frac{d^{2}}{d x^{2}} \Im{x}\right) \sin{\left (x \right )} - \cos{\left (x \right )}\right)$$