Производная (Abs(cos(x)))

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↑ Функция f () ? - производная -го порядка

Решение

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[LaTeX]
|cos(x)|
$$\left|{\cos{\left (x \right )}}\right|$$
Первая производная
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/d                                  d                               \                          /  d                                  d                               \                       
|--(im(x))*cos(re(x))*sinh(im(x)) - --(re(x))*cosh(im(x))*sin(re(x))|*cos(re(x))*cosh(im(x)) - |- --(im(x))*cosh(im(x))*sin(re(x)) - --(re(x))*cos(re(x))*sinh(im(x))|*sin(re(x))*sinh(im(x))
\dx                                 dx                              /                          \  dx                                 dx                              /                       
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                                                                                           |cos(x)|                                                                                          
$$\frac{1}{\left|{\cos{\left (x \right )}}\right|} \left(\left(- \sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} + \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x}\right) \cos{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} - \left(- \sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x} - \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x}\right) \sin{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )}\right)$$
Вторая производная
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  //d                                  d                               \                          /d                                  d                               \                       \                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             
  ||--(re(x))*cosh(im(x))*sin(re(x)) - --(im(x))*cos(re(x))*sinh(im(x))|*cos(re(x))*cosh(im(x)) - |--(im(x))*cosh(im(x))*sin(re(x)) + --(re(x))*cos(re(x))*sinh(im(x))|*sin(re(x))*sinh(im(x))|    /           2                            2                                   2                                            2                                                                      \                          /           2                            2                                            2                            2                                                                             \                                                                                                                                                                                                                                                                                                                                                                                                                                                           
  \\dx                                 dx                              /                          \dx                                 dx                              /                       /    |/d        \                            d                                   d                                  /d        \                             d         d                               |                          |/d        \                            d                                  /d        \                            d                                    d         d                               |                          /d                                  d                               \ d                                  /d                                  d                               \ d                                  /d                                  d                               \ d                                  /d                                  d                               \ d                               
- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ||--(im(x))| *sin(re(x))*sinh(im(x)) + ---(im(x))*cosh(im(x))*sin(re(x)) + ---(re(x))*cos(re(x))*sinh(im(x)) - |--(re(x))| *sin(re(x))*sinh(im(x)) + 2*--(im(x))*--(re(x))*cos(re(x))*cosh(im(x))|*sin(re(x))*sinh(im(x)) - ||--(re(x))| *cos(re(x))*cosh(im(x)) + ---(re(x))*cosh(im(x))*sin(re(x)) - |--(im(x))| *cos(re(x))*cosh(im(x)) - ---(im(x))*cos(re(x))*sinh(im(x)) + 2*--(im(x))*--(re(x))*sin(re(x))*sinh(im(x))|*cos(re(x))*cosh(im(x)) + |--(im(x))*cosh(im(x))*sin(re(x)) + --(re(x))*cos(re(x))*sinh(im(x))|*--(im(x))*cosh(im(x))*sin(re(x)) + |--(im(x))*cosh(im(x))*sin(re(x)) + --(re(x))*cos(re(x))*sinh(im(x))|*--(re(x))*cos(re(x))*sinh(im(x)) + |--(re(x))*cosh(im(x))*sin(re(x)) - --(im(x))*cos(re(x))*sinh(im(x))|*--(re(x))*cosh(im(x))*sin(re(x)) - |--(re(x))*cosh(im(x))*sin(re(x)) - --(im(x))*cos(re(x))*sinh(im(x))|*--(im(x))*cos(re(x))*sinh(im(x))
                                                                                                    2                                                                                              |\dx       /                             2                                   2                                 \dx       /                             dx        dx                              |                          |\dx       /                             2                                 \dx       /                             2                                   dx        dx                              |                          \dx                                 dx                              / dx                                 \dx                                 dx                              / dx                                 \dx                                 dx                              / dx                                 \dx                                 dx                              / dx                              
                                                                                            |cos(x)|                                                                                               \                                      dx                                  dx                                                                                                                    /                          \                                      dx                                                                        dx                                                                              /                                                                                                                                                                                                                                                                                                                                                                                                                                                           
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                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |cos(x)|                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          
$$\frac{1}{\left|{\cos{\left (x \right )}}\right|} \left(- \frac{1}{\left|{\cos{\left (x \right )}}\right|^{2}} \left(\left(\sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} - \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x}\right) \cos{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} - \left(\sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x} + \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x}\right) \sin{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )}\right)^{2} + \left(\sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} - \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x}\right) \sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} - \left(\sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} - \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x}\right) \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x} + \left(\sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x} + \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x}\right) \sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x} + \left(\sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x} + \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x}\right) \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} + \left(- \sin{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x}^{2} + \sin{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x}^{2} + \sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d^{2}}{d x^{2}} \Im{x} + \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d^{2}}{d x^{2}} \Re{x} + 2 \cos{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} \frac{d}{d x} \Im{x}\right) \sin{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} - \left(2 \sin{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} \frac{d}{d x} \Im{x} + \sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d^{2}}{d x^{2}} \Re{x} - \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d^{2}}{d x^{2}} \Im{x} + \cos{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x}^{2} - \cos{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x}^{2}\right) \cos{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )}\right)$$
Третья производная
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                                                                                                                                                                                                 3                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       //d                                  d                               \                          /d                                  d                               \                       \ ||/d        \                            d                                   d                                  /d        \                             d         d                               |                          |/d        \                            d                                  /d        \                            d                                    d         d                               |                          /d                                  d                               \ d                                  /d                                  d                               \ d                                  /d                                  d                               \ d                                  /d                                  d                               \ d                               |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  
    //d                                  d                               \                          /d                                  d                               \                       \                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      3*||--(re(x))*cosh(im(x))*sin(re(x)) - --(im(x))*cos(re(x))*sinh(im(x))|*cos(re(x))*cosh(im(x)) - |--(im(x))*cosh(im(x))*sin(re(x)) + --(re(x))*cos(re(x))*sinh(im(x))|*sin(re(x))*sinh(im(x))|*|||--(im(x))| *sin(re(x))*sinh(im(x)) + ---(im(x))*cosh(im(x))*sin(re(x)) + ---(re(x))*cos(re(x))*sinh(im(x)) - |--(re(x))| *sin(re(x))*sinh(im(x)) + 2*--(im(x))*--(re(x))*cos(re(x))*cosh(im(x))|*sin(re(x))*sinh(im(x)) - ||--(re(x))| *cos(re(x))*cosh(im(x)) + ---(re(x))*cosh(im(x))*sin(re(x)) - |--(im(x))| *cos(re(x))*cosh(im(x)) - ---(im(x))*cos(re(x))*sinh(im(x)) + 2*--(im(x))*--(re(x))*sin(re(x))*sinh(im(x))|*cos(re(x))*cosh(im(x)) + |--(im(x))*cosh(im(x))*sin(re(x)) + --(re(x))*cos(re(x))*sinh(im(x))|*--(im(x))*cosh(im(x))*sin(re(x)) + |--(im(x))*cosh(im(x))*sin(re(x)) + --(re(x))*cos(re(x))*sinh(im(x))|*--(re(x))*cos(re(x))*sinh(im(x)) + |--(re(x))*cosh(im(x))*sin(re(x)) - --(im(x))*cos(re(x))*sinh(im(x))|*--(re(x))*cosh(im(x))*sin(re(x)) - |--(re(x))*cosh(im(x))*sin(re(x)) - --(im(x))*cos(re(x))*sinh(im(x))|*--(im(x))*cos(re(x))*sinh(im(x))|                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  
  3*||--(re(x))*cosh(im(x))*sin(re(x)) - --(im(x))*cos(re(x))*sinh(im(x))|*cos(re(x))*cosh(im(x)) - |--(im(x))*cosh(im(x))*sin(re(x)) + --(re(x))*cos(re(x))*sinh(im(x))|*sin(re(x))*sinh(im(x))|    /           3                            3                                   3                                            3                                       2                                                  2                                              2                                                  2                                               2                                               2                              \                          /  3                                            3                                     3                            3                                               2                                              2                                                 2                                                  2                                               2                                               2                              \                            \\dx                                 dx                              /                          \dx                                 dx                              /                       / ||\dx       /                             2                                   2                                 \dx       /                             dx        dx                              |                          |\dx       /                             2                                 \dx       /                             2                                   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                                                                      \                                      /           2                            2                                   2                                            2                                                                      \                                      /           2                            2                                            2                            2                                                                             \                                                                                                                                                                                                                                                                           
    \\dx                                 dx                              /                          \dx                                 dx                              /                       /    |/d        \                            d                                   d                                  /d        \                             /d        \  d                                    d          d                                    /d        \  d                                    d          d                                    d          d                                    d          d                               |                          | d                                  /d        \                           /d        \                            d                                    d          d                                    /d        \  d                                    /d        \  d                                    d          d                                    d          d                                    d          d                               |                                                                                                                                                                                                                          \\                                      dx                                  dx                                                                                                                    /                          \                                      dx                                                                        dx                                                                              /                                                                                                                                                                                                                                                                                                                                                                                                                                                           /   /d        \  /d                                  d                               \                          /d        \  /d                                  d                               \                          /d                                  d                               \  d                                  /d                                  d                               \  d                                  /d                                  d                               \  d                                  /d        \  /d                                  d                               \                          /d        \  /d                                  d                               \                          /d                                  d                               \  d                                    |/d        \                            d                                  /d        \                            d                                    d         d                               | d                                    |/d        \                            d                                   d                                  /d        \                             d         d                               | d                                    |/d        \                            d                                   d                                  /d        \                             d         d                               | d                                    |/d        \                            d                                  /d        \                            d                                    d         d                               | d                                    /d                                  d                               \ d         d                                    /d                                  d                               \ d         d                               
- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ + ||--(im(x))| *cosh(im(x))*sin(re(x)) + ---(im(x))*cosh(im(x))*sin(re(x)) + ---(re(x))*cos(re(x))*sinh(im(x)) - |--(re(x))| *cos(re(x))*sinh(im(x)) - 3*|--(re(x))| *--(im(x))*cosh(im(x))*sin(re(x)) - 3*--(re(x))*---(re(x))*sin(re(x))*sinh(im(x)) + 3*|--(im(x))| *--(re(x))*cos(re(x))*sinh(im(x)) + 3*--(im(x))*---(im(x))*sin(re(x))*sinh(im(x)) + 3*--(im(x))*---(re(x))*cos(re(x))*cosh(im(x)) + 3*--(re(x))*---(im(x))*cos(re(x))*cosh(im(x))|*sin(re(x))*sinh(im(x)) - |---(re(x))*cosh(im(x))*sin(re(x)) - |--(im(x))| *cos(re(x))*sinh(im(x)) - |--(re(x))| *cosh(im(x))*sin(re(x)) - ---(im(x))*cos(re(x))*sinh(im(x)) - 3*--(im(x))*---(im(x))*cos(re(x))*cosh(im(x)) + 3*|--(im(x))| *--(re(x))*cosh(im(x))*sin(re(x)) + 3*|--(re(x))| *--(im(x))*cos(re(x))*sinh(im(x)) + 3*--(im(x))*---(re(x))*sin(re(x))*sinh(im(x)) + 3*--(re(x))*---(im(x))*sin(re(x))*sinh(im(x)) + 3*--(re(x))*---(re(x))*cos(re(x))*cosh(im(x))|*cos(re(x))*cosh(im(x)) + ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + |--(im(x))| *|--(im(x))*cosh(im(x))*sin(re(x)) + --(re(x))*cos(re(x))*sinh(im(x))|*sin(re(x))*sinh(im(x)) + |--(re(x))| *|--(re(x))*cosh(im(x))*sin(re(x)) - --(im(x))*cos(re(x))*sinh(im(x))|*cos(re(x))*cosh(im(x)) + |--(im(x))*cosh(im(x))*sin(re(x)) + --(re(x))*cos(re(x))*sinh(im(x))|*---(im(x))*cosh(im(x))*sin(re(x)) + |--(im(x))*cosh(im(x))*sin(re(x)) + --(re(x))*cos(re(x))*sinh(im(x))|*---(re(x))*cos(re(x))*sinh(im(x)) + |--(re(x))*cosh(im(x))*sin(re(x)) - --(im(x))*cos(re(x))*sinh(im(x))|*---(re(x))*cosh(im(x))*sin(re(x)) - |--(im(x))| *|--(re(x))*cosh(im(x))*sin(re(x)) - --(im(x))*cos(re(x))*sinh(im(x))|*cos(re(x))*cosh(im(x)) - |--(re(x))| *|--(im(x))*cosh(im(x))*sin(re(x)) + --(re(x))*cos(re(x))*sinh(im(x))|*sin(re(x))*sinh(im(x)) - |--(re(x))*cosh(im(x))*sin(re(x)) - --(im(x))*cos(re(x))*sinh(im(x))|*---(im(x))*cos(re(x))*sinh(im(x)) - 2*||--(re(x))| *cos(re(x))*cosh(im(x)) + ---(re(x))*cosh(im(x))*sin(re(x)) - |--(im(x))| *cos(re(x))*cosh(im(x)) - ---(im(x))*cos(re(x))*sinh(im(x)) + 2*--(im(x))*--(re(x))*sin(re(x))*sinh(im(x))|*--(im(x))*cos(re(x))*sinh(im(x)) + 2*||--(im(x))| *sin(re(x))*sinh(im(x)) + ---(im(x))*cosh(im(x))*sin(re(x)) + ---(re(x))*cos(re(x))*sinh(im(x)) - |--(re(x))| *sin(re(x))*sinh(im(x)) + 2*--(im(x))*--(re(x))*cos(re(x))*cosh(im(x))|*--(im(x))*cosh(im(x))*sin(re(x)) + 2*||--(im(x))| *sin(re(x))*sinh(im(x)) + ---(im(x))*cosh(im(x))*sin(re(x)) + ---(re(x))*cos(re(x))*sinh(im(x)) - |--(re(x))| *sin(re(x))*sinh(im(x)) + 2*--(im(x))*--(re(x))*cos(re(x))*cosh(im(x))|*--(re(x))*cos(re(x))*sinh(im(x)) + 2*||--(re(x))| *cos(re(x))*cosh(im(x)) + ---(re(x))*cosh(im(x))*sin(re(x)) - |--(im(x))| *cos(re(x))*cosh(im(x)) - ---(im(x))*cos(re(x))*sinh(im(x)) + 2*--(im(x))*--(re(x))*sin(re(x))*sinh(im(x))|*--(re(x))*cosh(im(x))*sin(re(x)) + 2*|--(im(x))*cosh(im(x))*sin(re(x)) + --(re(x))*cos(re(x))*sinh(im(x))|*--(im(x))*--(re(x))*cos(re(x))*cosh(im(x)) + 2*|--(re(x))*cosh(im(x))*sin(re(x)) - --(im(x))*cos(re(x))*sinh(im(x))|*--(im(x))*--(re(x))*sin(re(x))*sinh(im(x))
                                                                                                     4                                                                                               |\dx       /                             3                                   3                                 \dx       /                             \dx       /  dx                                   dx          2                                   \dx       /  dx                                   dx          2                                   dx          2                                   dx          2                              |                          |  3                                 \dx       /                           \dx       /                             3                                   dx          2                                   \dx       /  dx                                   \dx       /  dx                                   dx          2                                   dx          2                                   dx          2                              |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           2                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            \dx       /  \dx                                 dx                              /                          \dx       /  \dx                                 dx                              /                          \dx                                 dx                              /   2                                 \dx                                 dx                              /   2                                 \dx                                 dx                              /   2                                 \dx       /  \dx                                 dx                              /                          \dx       /  \dx                                 dx                              /                          \dx                                 dx                              /   2                                   |\dx       /                             2                                 \dx       /                             2                                   dx        dx                              | dx                                   |\dx       /                             2                                   2                                 \dx       /                             dx        dx                              | dx                                   |\dx       /                             2                                   2                                 \dx       /                             dx        dx                              | dx                                   |\dx       /                             2                                 \dx       /                             2                                   dx        dx                              | dx                                   \dx                                 dx                              / dx        dx                                   \dx                                 dx                              / dx        dx                              
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$$\frac{1}{\left|{\cos{\left (x \right )}}\right|} \left(- \frac{3}{\left|{\cos{\left (x \right )}}\right|^{4}} \left(\left(\sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} - \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x}\right) \cos{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} - \left(\sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x} + \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x}\right) \sin{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )}\right)^{3} + \frac{3}{\left|{\cos{\left (x \right )}}\right|^{2}} \left(\left(\sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} - \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x}\right) \cos{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} - \left(\sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x} + \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x}\right) \sin{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )}\right) \left(\left(\sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} - \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x}\right) \sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} - \left(\sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} - \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x}\right) \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x} + \left(\sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x} + \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x}\right) \sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x} + \left(\sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x} + \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x}\right) \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} + \left(- \sin{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x}^{2} + \sin{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x}^{2} + \sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d^{2}}{d x^{2}} \Im{x} + \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d^{2}}{d x^{2}} \Re{x} + 2 \cos{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} \frac{d}{d x} \Im{x}\right) \sin{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} - \left(2 \sin{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} \frac{d}{d x} \Im{x} + \sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d^{2}}{d x^{2}} \Re{x} - \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d^{2}}{d x^{2}} \Im{x} + \cos{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x}^{2} - \cos{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x}^{2}\right) \cos{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )}\right) + 2 \left(\sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} - \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x}\right) \sin{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} \frac{d}{d x} \Im{x} + \left(\sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} - \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x}\right) \sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d^{2}}{d x^{2}} \Re{x} - \left(\sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} - \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x}\right) \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d^{2}}{d x^{2}} \Im{x} + \left(\sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} - \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x}\right) \cos{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x}^{2} - \left(\sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} - \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x}\right) \cos{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x}^{2} - \left(\sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x} + \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x}\right) \sin{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x}^{2} + \left(\sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x} + \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x}\right) \sin{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x}^{2} + \left(\sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x} + \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x}\right) \sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d^{2}}{d x^{2}} \Im{x} + \left(\sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x} + \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x}\right) \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d^{2}}{d x^{2}} \Re{x} + 2 \left(\sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x} + \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x}\right) \cos{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} \frac{d}{d x} \Im{x} + 2 \left(- \sin{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x}^{2} + \sin{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x}^{2} + \sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d^{2}}{d x^{2}} \Im{x} + \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d^{2}}{d x^{2}} \Re{x} + 2 \cos{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} \frac{d}{d x} \Im{x}\right) \sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x} + 2 \left(- \sin{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x}^{2} + \sin{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x}^{2} + \sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d^{2}}{d x^{2}} \Im{x} + \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d^{2}}{d x^{2}} \Re{x} + 2 \cos{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} \frac{d}{d x} \Im{x}\right) \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} + 2 \left(2 \sin{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} \frac{d}{d x} \Im{x} + \sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d^{2}}{d x^{2}} \Re{x} - \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d^{2}}{d x^{2}} \Im{x} + \cos{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x}^{2} - \cos{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x}^{2}\right) \sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} - 2 \left(2 \sin{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} \frac{d}{d x} \Im{x} + \sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d^{2}}{d x^{2}} \Re{x} - \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d^{2}}{d x^{2}} \Im{x} + \cos{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x}^{2} - \cos{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x}^{2}\right) \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x} + \left(- 3 \sin{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} \frac{d^{2}}{d x^{2}} \Re{x} + 3 \sin{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x} \frac{d^{2}}{d x^{2}} \Im{x} - 3 \sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x}^{2} \frac{d}{d x} \Im{x} + \sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x}^{3} + \sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d^{3}}{d x^{3}} \Im{x} - \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x}^{3} + 3 \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} \frac{d}{d x} \Im{x}^{2} + \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d^{3}}{d x^{3}} \Re{x} + 3 \cos{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} \frac{d^{2}}{d x^{2}} \Im{x} + 3 \cos{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x} \frac{d^{2}}{d x^{2}} \Re{x}\right) \sin{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} - \left(3 \sin{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} \frac{d^{2}}{d x^{2}} \Im{x} + 3 \sin{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x} \frac{d^{2}}{d x^{2}} \Re{x} - \sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x}^{3} + 3 \sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} \frac{d}{d x} \Im{x}^{2} + \sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d^{3}}{d x^{3}} \Re{x} + 3 \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x}^{2} \frac{d}{d x} \Im{x} - \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x}^{3} - \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d^{3}}{d x^{3}} \Im{x} + 3 \cos{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} \frac{d^{2}}{d x^{2}} \Re{x} - 3 \cos{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x} \frac{d^{2}}{d x^{2}} \Im{x}\right) \cos{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )}\right)$$