Производная Abs(log(x))

Вы ввели

|log(x)|

$$\left|{\log{\left (x \right )}}\right|$$

Первая производная

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

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

Вторая производная

                                                                                                                                                                         2                                                                                                                                                                                                                                                                   
                                                                              //d                 d              \          /d                 d              \         \                                                                                                                                                                                                                                                                    
                                                                              ||--(im(x))*re(x) - --(re(x))*im(x)|*arg(x)   |--(im(x))*im(x) + --(re(x))*re(x)|*log(|x|)|    /  2                  2             \          /           2              2     2                  2             \                                                                                                                                              
                                   2                                      2   |\dx                dx             /          \dx                dx             /         |    | d                  d              |          |/d        \    /d        \     d                  d              |                                                 2                                                                                            
/d                 d              \    /d                 d              \    |------------------------------------------ + --------------------------------------------|    |---(im(x))*re(x) - ---(re(x))*im(x)|*arg(x)   ||--(im(x))|  + |--(re(x))|  + ---(im(x))*im(x) + ---(re(x))*re(x)|*log(|x|)     /d                 d              \               /d                 d              \ /d                 d              \       
|--(im(x))*re(x) - --(re(x))*im(x)|    |--(im(x))*im(x) + --(re(x))*re(x)|    |               2        2                                           2                    |    |  2                  2             |          |\dx       /    \dx       /      2                  2             |            2*|--(im(x))*im(x) + --(re(x))*re(x)| *log(|x|)   2*|--(im(x))*im(x) + --(re(x))*re(x)|*|--(im(x))*re(x) - --(re(x))*im(x)|*arg(x)
\dx                dx             /    \dx                dx             /    \             im (x) + re (x)                                     |x|                     /    \dx                 dx              /          \                              dx                 dx              /              \dx                dx             /               \dx                dx             / \dx                dx             /       
------------------------------------ + ------------------------------------ - -------------------------------------------------------------------------------------------- + -------------------------------------------- + ---------------------------------------------------------------------------- - ----------------------------------------------- - --------------------------------------------------------------------------------
                          2                               4                                                                    2                                                             2        2                                                            2                                                                4                                                                        2                               
         /  2        2   \                             |x|                                                             |log(x)|                                                            im (x) + re (x)                                                      |x|                                                              |x|                                                        /  2        2   \                                
         \im (x) + re (x)/                                                                                                                                                                                                                                                                                                                                                                  \im (x) + re (x)/                                
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                                                                                                                                                                                                                   |log(x)|                                                                                                                                                                                                                  

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

Третья производная

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       /                                                                              /  2                  2             \          /           2              2     2                  2             \                                                                                                                                              \                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       |                                   2                                      2   | d                  d              |          |/d        \    /d        \     d                  d              |                                                 2                                                                                            |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             
                                                                                                                                        3                                                                                                                                                                                                                                                                                  //d                 d              \          /d                 d              \         \ |/d                 d              \    /d                 d              \    |---(im(x))*re(x) - ---(re(x))*im(x)|*arg(x)   ||--(im(x))|  + |--(re(x))|  + ---(im(x))*im(x) + ---(re(x))*re(x)|*log(|x|)     /d                 d              \               /d                 d              \ /d                 d              \       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             
                                             //d                 d              \          /d                 d              \         \                                                                                                                                                                                                                                                                                   ||--(im(x))*re(x) - --(re(x))*im(x)|*arg(x)   |--(im(x))*im(x) + --(re(x))*re(x)|*log(|x|)| ||--(im(x))*re(x) - --(re(x))*im(x)|    |--(im(x))*im(x) + --(re(x))*re(x)|    |  2                  2             |          |\dx       /    \dx       /      2                  2             |            2*|--(im(x))*im(x) + --(re(x))*re(x)| *log(|x|)   2*|--(im(x))*im(x) + --(re(x))*re(x)|*|--(im(x))*re(x) - --(re(x))*im(x)|*arg(x)|                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             
                                             ||--(im(x))*re(x) - --(re(x))*im(x)|*arg(x)   |--(im(x))*im(x) + --(re(x))*re(x)|*log(|x|)|    /            2            3                            2            3             \          /  3                  3                              2                        2       \                                                                                           |\dx                dx             /          \dx                dx             /         | |\dx                dx             /    \dx                dx             /    \dx                 dx              /          \                              dx                 dx              /              \dx                dx             /               \dx                dx             / \dx                dx             /       |                                         /  2                  2             \                                         /           2              2     2                  2             \                                                                                           /           2              2     2                  2             \                                                  /  2                  2             \                                                /           2              2     2                  2             \                                                                                           
                                       3     |\dx                dx             /          \dx                dx             /         |    |d          d            d                 d          d            d              |          | d                  d                   d          d             d          d        |                                                 2                                       3*|------------------------------------------ + --------------------------------------------|*|------------------------------------ + ------------------------------------ + -------------------------------------------- + ---------------------------------------------------------------------------- - ----------------------------------------------- - --------------------------------------------------------------------------------|     /d                 d              \ | d                  d              |     /d                 d              \ |/d        \    /d        \     d                  d              |                                        3              /d                 d              \ |/d        \    /d        \     d                  d              |              /d                 d              \ | d                  d              |            /d                 d              \ |/d        \    /d        \     d                  d              |                                               2                                           
    /d                 d              \    3*|------------------------------------------ + --------------------------------------------|    |--(re(x))*---(im(x)) + ---(im(x))*re(x) - --(im(x))*---(re(x)) - ---(re(x))*im(x)|*arg(x)   |---(im(x))*im(x) + ---(re(x))*re(x) + 3*--(im(x))*---(im(x)) + 3*--(re(x))*---(re(x))|*log(|x|)     /d                 d              \  /d                 d              \     |               2        2                                           2                    | |                          2                               4                                   2        2                                                            2                                                                4                                                                        2                               |   3*|--(im(x))*re(x) - --(re(x))*im(x)|*|---(im(x))*re(x) - ---(re(x))*im(x)|   3*|--(im(x))*im(x) + --(re(x))*re(x)|*||--(im(x))|  + |--(re(x))|  + ---(im(x))*im(x) + ---(re(x))*re(x)|     /d                 d              \             6*|--(im(x))*im(x) + --(re(x))*re(x)|*||--(im(x))|  + |--(re(x))|  + ---(im(x))*im(x) + ---(re(x))*re(x)|*log(|x|)   4*|--(im(x))*im(x) + --(re(x))*re(x)|*|---(im(x))*re(x) - ---(re(x))*im(x)|*arg(x)   2*|--(im(x))*re(x) - --(re(x))*im(x)|*||--(im(x))|  + |--(re(x))|  + ---(im(x))*im(x) + ---(re(x))*re(x)|*arg(x)     /d                 d              \  /d                 d              \       
  6*|--(im(x))*im(x) + --(re(x))*re(x)|      |               2        2                                           2                    |    |dx          2            3                dx          2            3             |          |  3                  3                  dx          2            dx          2       |            6*|--(im(x))*re(x) - --(re(x))*im(x)| *|--(im(x))*im(x) + --(re(x))*re(x)|     \             im (x) + re (x)                                     |x|                     / |         /  2        2   \                             |x|                                  im (x) + re (x)                                                      |x|                                                              |x|                                                        /  2        2   \                                |     \dx                dx             / |  2                  2             |     \dx                dx             / |\dx       /    \dx       /      2                  2             |   8*|--(im(x))*im(x) + --(re(x))*re(x)| *log(|x|)     \dx                dx             / |\dx       /    \dx       /      2                  2             |              \dx                dx             / |  2                  2             |            \dx                dx             / |\dx       /    \dx       /      2                  2             |          8*|--(im(x))*im(x) + --(re(x))*re(x)| *|--(im(x))*re(x) - --(re(x))*im(x)|*arg(x)
    \dx                dx             /      \             im (x) + re (x)                                     |x|                     /    \          dx           dx                           dx           dx              /          \dx                 dx                             dx                       dx        /              \dx                dx             /  \dx                dx             /                                                                                                 \         \im (x) + re (x)/                                                                                                                                                                                                                                                                   \im (x) + re (x)/                                /                                         \dx                 dx              /                                         \                              dx                 dx              /     \dx                dx             /                                                   \                              dx                 dx              /                                                  \dx                 dx              /                                                \                              dx                 dx              /            \dx                dx             /  \dx                dx             /       
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                      6                                                                      4                                                                                     2        2                                                                                             2                                                                                              3                                                                                                                                                                                                                                                         2                                                                                                                                                                                                                                                                    2                                                                                     4                                                                              6                                                                                  4                                                                                                           2                                                                                                   2                                                                                                   3                               
                   |x|                                                               |log(x)|                                                                                    im (x) + re (x)                                                                                       |x|                                                                              /  2        2   \                                                                                                                                                                                                                                                  |log(x)|                                                                                                                                                                                                                                                    /  2        2   \                                                                                   |x|                                                                            |x|                                                                                |x|                                                                                           /  2        2   \                                                                                   /  2        2   \                                                                                   /  2        2   \                                
                                                                                                                                                                                                                                                                                                                                                                        \im (x) + re (x)/                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              \im (x) + re (x)/                                                                                                                                                                                                                                                                                                                                                   \im (x) + re (x)/                                                                                   \im (x) + re (x)/                                                                                   \im (x) + re (x)/                                
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                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              |log(x)|                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              

$$\frac{1}{\left|{\log{\left (x \right )}}\right|} \left(\frac{8}{\left|{x}\right|^{6}} \left(\Re{x} \frac{d}{d x} \Re{x} + \Im{x} \frac{d}{d x} \Im{x}\right)^{3} \log{\left (\left|{x}\right| \right )} - \frac{6}{\left|{x}\right|^{6}} \left(\Re{x} \frac{d}{d x} \Re{x} + \Im{x} \frac{d}{d x} \Im{x}\right)^{3} + \frac{8 \arg{\left (x \right )}}{\left(\left(\Re{x}\right)^{2} + \left(\Im{x}\right)^{2}\right)^{3}} \left(\Re{x} \frac{d}{d x} \Re{x} + \Im{x} \frac{d}{d x} \Im{x}\right)^{2} \left(\Re{x} \frac{d}{d x} \Im{x} - \Im{x} \frac{d}{d x} \Re{x}\right) - \frac{6}{\left(\left(\Re{x}\right)^{2} + \left(\Im{x}\right)^{2}\right)^{3}} \left(\Re{x} \frac{d}{d x} \Re{x} + \Im{x} \frac{d}{d x} \Im{x}\right) \left(\Re{x} \frac{d}{d x} \Im{x} - \Im{x} \frac{d}{d x} \Re{x}\right)^{2} - \frac{4 \arg{\left (x \right )}}{\left(\left(\Re{x}\right)^{2} + \left(\Im{x}\right)^{2}\right)^{2}} \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}} \Im{x} - \Im{x} \frac{d^{2}}{d x^{2}} \Re{x}\right) - \frac{6}{\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) \log{\left (\left|{x}\right| \right )} + \frac{3}{\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) + \frac{3}{\left(\left(\Re{x}\right)^{2} + \left(\Im{x}\right)^{2}\right)^{2}} \left(\Re{x} \frac{d}{d x} \Im{x} - \Im{x} \frac{d}{d x} \Re{x}\right) \left(\Re{x} \frac{d^{2}}{d x^{2}} \Im{x} - \Im{x} \frac{d^{2}}{d x^{2}} \Re{x}\right) - \frac{2 \arg{\left (x \right )}}{\left(\left(\Re{x}\right)^{2} + \left(\Im{x}\right)^{2}\right)^{2}} \left(\Re{x} \frac{d}{d x} \Im{x} - \Im{x} \frac{d}{d x} \Re{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) + \frac{3}{\left|{\log{\left (x \right )}}\right|^{4}} \left(\frac{1}{\left|{x}\right|^{2}} \left(\Re{x} \frac{d}{d x} \Re{x} + \Im{x} \frac{d}{d x} \Im{x}\right) \log{\left (\left|{x}\right| \right )} + \frac{\arg{\left (x \right )}}{\left(\Re{x}\right)^{2} + \left(\Im{x}\right)^{2}} \left(\Re{x} \frac{d}{d x} \Im{x} - \Im{x} \frac{d}{d x} \Re{x}\right)\right)^{3} - \frac{3}{\left|{\log{\left (x \right )}}\right|^{2}} \left(\frac{1}{\left|{x}\right|^{2}} \left(\Re{x} \frac{d}{d x} \Re{x} + \Im{x} \frac{d}{d x} \Im{x}\right) \log{\left (\left|{x}\right| \right )} + \frac{\arg{\left (x \right )}}{\left(\Re{x}\right)^{2} + \left(\Im{x}\right)^{2}} \left(\Re{x} \frac{d}{d x} \Im{x} - \Im{x} \frac{d}{d x} \Re{x}\right)\right) \left(- \frac{2}{\left|{x}\right|^{4}} \left(\Re{x} \frac{d}{d x} \Re{x} + \Im{x} \frac{d}{d x} \Im{x}\right)^{2} \log{\left (\left|{x}\right| \right )} + \frac{1}{\left|{x}\right|^{4}} \left(\Re{x} \frac{d}{d x} \Re{x} + \Im{x} \frac{d}{d x} \Im{x}\right)^{2} - \frac{2 \arg{\left (x \right )}}{\left(\left(\Re{x}\right)^{2} + \left(\Im{x}\right)^{2}\right)^{2}} \left(\Re{x} \frac{d}{d x} \Re{x} + \Im{x} \frac{d}{d x} \Im{x}\right) \left(\Re{x} \frac{d}{d x} \Im{x} - \Im{x} \frac{d}{d x} \Re{x}\right) + \frac{1}{\left(\left(\Re{x}\right)^{2} + \left(\Im{x}\right)^{2}\right)^{2}} \left(\Re{x} \frac{d}{d x} \Im{x} - \Im{x} \frac{d}{d x} \Re{x}\right)^{2} + \frac{\arg{\left (x \right )}}{\left(\Re{x}\right)^{2} + \left(\Im{x}\right)^{2}} \left(\Re{x} \frac{d^{2}}{d x^{2}} \Im{x} - \Im{x} \frac{d^{2}}{d x^{2}} \Re{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) \log{\left (\left|{x}\right| \right )}\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) \log{\left (\left|{x}\right| \right )} + \frac{\arg{\left (x \right )}}{\left(\Re{x}\right)^{2} + \left(\Im{x}\right)^{2}} \left(\Re{x} \frac{d^{3}}{d x^{3}} \Im{x} - \Im{x} \frac{d^{3}}{d x^{3}} \Re{x} + \frac{d}{d x} \Re{x} \frac{d^{2}}{d x^{2}} \Im{x} - \frac{d}{d x} \Im{x} \frac{d^{2}}{d x^{2}} \Re{x}\right)\right)$$