Производная (|x+1|)

Вы ввели

|x + 1|

$$\left|{x + 1}\right|$$

Первая производная

            d           d              
(1 + re(x))*--(re(x)) + --(im(x))*im(x)
            dx          dx             
---------------------------------------
                |x + 1|                

$$\frac{1}{\left|{x + 1}\right|} \left(\left(\Re{x} + 1\right) \frac{d}{d x} \Re{x} + \Im{x} \frac{d}{d x} \Im{x}\right)$$

Вторая производная

                                                                                                                   2
                                                                          /            d           d              \ 
           2              2                 2            2                |(1 + re(x))*--(re(x)) + --(im(x))*im(x)| 
/d        \    /d        \                 d            d                 \            dx          dx             / 
|--(im(x))|  + |--(re(x))|  + (1 + re(x))*---(re(x)) + ---(im(x))*im(x) - ------------------------------------------
\dx       /    \dx       /                  2            2                                        2                 
                                          dx           dx                                  |1 + x|                  
--------------------------------------------------------------------------------------------------------------------
                                                      |1 + x|                                                       

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

Третья производная

                                                                                                                                                                                         /           2              2                 2            2             \
                                                                                       3                                                       /            d           d              \ |/d        \    /d        \                 d            d              |
                                              /            d           d              \                                                      3*|(1 + re(x))*--(re(x)) + --(im(x))*im(x)|*||--(im(x))|  + |--(re(x))|  + (1 + re(x))*---(re(x)) + ---(im(x))*im(x)|
              3            3                3*|(1 + re(x))*--(re(x)) + --(im(x))*im(x)|                  2                        2            \            dx          dx             / |\dx       /    \dx       /                  2            2             |
             d            d                   \            dx          dx             /      d          d             d          d                                                       \                                          dx           dx              /
(1 + re(x))*---(re(x)) + ---(im(x))*im(x) + -------------------------------------------- + 3*--(im(x))*---(im(x)) + 3*--(re(x))*---(re(x)) - ---------------------------------------------------------------------------------------------------------------------
              3            3                                         4                       dx          2            dx          2                                                                        2                                                      
            dx           dx                                   |1 + x|                                  dx                       dx                                                                  |1 + x|                                                       
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                             |1 + x|                                                                                                                              

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

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