Производная функции/ От одной переменной/ y' = ((|x^3|))'

Производная (|x^3|)

Учитель очень удивится увидев твоё верное решение производной 😼

()'

– производная -го порядка в точке

Примеры

График:

от до

Кусочно-заданная:

{ кусочно-заданную функцию ввести здесь.

Решение

Вы ввели [src]
| 3|
|x |
$$\left|{x^{3}}\right|$$
Первая производная [src]
/  3          2         \ /      2    d               2    d             d                    \   /    3          2         \ /      2    d               2    d             d                    \
\re (x) - 3*im (x)*re(x)/*|- 3*im (x)*--(re(x)) + 3*re (x)*--(re(x)) - 6*--(im(x))*im(x)*re(x)| + \- im (x) + 3*re (x)*im(x)/*|- 3*im (x)*--(im(x)) + 3*re (x)*--(im(x)) + 6*--(re(x))*im(x)*re(x)|
                          \           dx                   dx            dx                   /                               \           dx                   dx            dx                   /
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                                                                                                | 3|                                                                                               
                                                                                                |x |
$$\frac{1}{\left|{x^{3}}\right|} \left(\left(3 \left(\Re{x}\right)^{2} \Im{x} - \left(\Im{x}\right)^{3}\right) \left(3 \left(\Re{x}\right)^{2} \frac{d}{d x} \Im{x} + 6 \Re{x} \Im{x} \frac{d}{d x} \Re{x} - 3 \left(\Im{x}\right)^{2} \frac{d}{d x} \Im{x}\right) + \left(\left(\Re{x}\right)^{3} - 3 \Re{x} \left(\Im{x}\right)^{2}\right) \left(3 \left(\Re{x}\right)^{2} \frac{d}{d x} \Re{x} - 6 \Re{x} \Im{x} \frac{d}{d x} \Im{x} - 3 \left(\Im{x}\right)^{2} \frac{d}{d x} \Re{x}\right)\right)$$
Упростить
Вторая производная [src]
  /                                                                                                                                                                                                                                                                                                                                     2                                                                                                                                                                                                                                                                                                                                                        \
  |                                                                                                                                            //    2          2   \ /  2    d             2    d             d                    \         /  2          2   \ /  2    d             2    d             d                    \      \                                                                                                                                                                                                                                                                                                                                                         |
  |                                                                 2                                                                    2   3*|\- im (x) + 3*re (x)/*|re (x)*--(im(x)) - im (x)*--(im(x)) + 2*--(re(x))*im(x)*re(x)|*im(x) - \re (x) - 3*im (x)/*|im (x)*--(re(x)) - re (x)*--(re(x)) + 2*--(im(x))*im(x)*re(x)|*re(x)|                          /         2                   2                       2                      2             2                                                 \                             /         2                   2                       2                      2             2                                                 \      |
  |  /  2    d             2    d             d                    \      /  2    d             2    d             d                    \      \                      \       dx                 dx            dx                   /                             \       dx                 dx            dx                   /      /    /    2          2   \ |  2     d             2     d             /d        \            /d        \             d                         d         d              |         /  2          2   \ |  2     d             2     d             /d        \            /d        \             d                         d         d              |      |
3*|3*|im (x)*--(re(x)) - re (x)*--(re(x)) + 2*--(im(x))*im(x)*re(x)|  + 3*|re (x)*--(im(x)) - im (x)*--(im(x)) + 2*--(re(x))*im(x)*re(x)|  - -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + \- im (x) + 3*re (x)/*|re (x)*---(im(x)) - im (x)*---(im(x)) - 2*|--(im(x))| *im(x) + 2*|--(re(x))| *im(x) + 2*---(re(x))*im(x)*re(x) + 4*--(im(x))*--(re(x))*re(x)|*im(x) - \re (x) - 3*im (x)/*|im (x)*---(re(x)) - re (x)*---(re(x)) - 2*|--(re(x))| *re(x) + 2*|--(im(x))| *re(x) + 2*---(im(x))*im(x)*re(x) + 4*--(im(x))*--(re(x))*im(x)|*re(x)|
  |  \       dx                 dx            dx                   /      \       dx                 dx            dx                   /                                                                                                   2                                                                                                                     |         2                   2            \dx       /            \dx       /              2                        dx        dx             |                             |         2                   2            \dx       /            \dx       /              2                        dx        dx             |      |
  |                                                                                                                                                                                                                                     | 3|                                                                                                                      \       dx                  dx                                                           dx                                                  /                             \       dx                  dx                                                           dx                                                  /      |
  \                                                                                                                                                                                                                                     |x |                                                                                                                                                                                                                                                                                                                                                                                                                                                     /
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                                                                                                                                                                                                                                                                                                                                               | 3|                                                                                                                                                                                                                                                                                                                                               
                                                                                                                                                                                                                                                                                                                                               |x |
$$\frac{1}{\left|{x^{3}}\right|} \left(- \frac{9}{\left|{x^{3}}\right|^{2}} \left(- \left(\left(\Re{x}\right)^{2} - 3 \left(\Im{x}\right)^{2}\right) \left(- \left(\Re{x}\right)^{2} \frac{d}{d x} \Re{x} + 2 \Re{x} \Im{x} \frac{d}{d x} \Im{x} + \left(\Im{x}\right)^{2} \frac{d}{d x} \Re{x}\right) \Re{x} + \left(3 \left(\Re{x}\right)^{2} - \left(\Im{x}\right)^{2}\right) \left(\left(\Re{x}\right)^{2} \frac{d}{d x} \Im{x} + 2 \Re{x} \Im{x} \frac{d}{d x} \Re{x} - \left(\Im{x}\right)^{2} \frac{d}{d x} \Im{x}\right) \Im{x}\right)^{2} - 3 \left(\left(\Re{x}\right)^{2} - 3 \left(\Im{x}\right)^{2}\right) \left(- \left(\Re{x}\right)^{2} \frac{d^{2}}{d x^{2}} \Re{x} + 2 \Re{x} \Im{x} \frac{d^{2}}{d x^{2}} \Im{x} - 2 \Re{x} \frac{d}{d x} \Re{x}^{2} + 2 \Re{x} \frac{d}{d x} \Im{x}^{2} + \left(\Im{x}\right)^{2} \frac{d^{2}}{d x^{2}} \Re{x} + 4 \Im{x} \frac{d}{d x} \Re{x} \frac{d}{d x} \Im{x}\right) \Re{x} + 3 \left(3 \left(\Re{x}\right)^{2} - \left(\Im{x}\right)^{2}\right) \left(\left(\Re{x}\right)^{2} \frac{d^{2}}{d x^{2}} \Im{x} + 2 \Re{x} \Im{x} \frac{d^{2}}{d x^{2}} \Re{x} + 4 \Re{x} \frac{d}{d x} \Re{x} \frac{d}{d x} \Im{x} - \left(\Im{x}\right)^{2} \frac{d^{2}}{d x^{2}} \Im{x} + 2 \Im{x} \frac{d}{d x} \Re{x}^{2} - 2 \Im{x} \frac{d}{d x} \Im{x}^{2}\right) \Im{x} + 9 \left(- \left(\Re{x}\right)^{2} \frac{d}{d x} \Re{x} + 2 \Re{x} \Im{x} \frac{d}{d x} \Im{x} + \left(\Im{x}\right)^{2} \frac{d}{d x} \Re{x}\right)^{2} + 9 \left(\left(\Re{x}\right)^{2} \frac{d}{d x} \Im{x} + 2 \Re{x} \Im{x} \frac{d}{d x} \Re{x} - \left(\Im{x}\right)^{2} \frac{d}{d x} \Im{x}\right)^{2}\right)$$
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Третья производная [src]
  /                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      /                                                                 2                                                                    2                         /         2                   2                       2                      2             2                                                 \                             /         2                   2                       2                      2             2                                                 \      \\
  |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  3                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         //    2          2   \ /  2    d             2    d             d                    \         /  2          2   \ /  2    d             2    d             d                    \      \ |  /  2    d             2    d             d                    \      /  2    d             2    d             d                    \    /    2          2   \ |  2     d             2     d             /d        \            /d        \             d                         d         d              |         /  2          2   \ |  2     d             2     d             /d        \            /d        \             d                         d         d              |      ||
  |                                                                                                                                                                                                                                                                                                                                                                                                                                         //    2          2   \ /  2    d             2    d             d                    \         /  2          2   \ /  2    d             2    d             d                    \      \                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        9*|\- im (x) + 3*re (x)/*|re (x)*--(im(x)) - im (x)*--(im(x)) + 2*--(re(x))*im(x)*re(x)|*im(x) - \re (x) - 3*im (x)/*|im (x)*--(re(x)) - re (x)*--(re(x)) + 2*--(im(x))*im(x)*re(x)|*re(x)|*|3*|im (x)*--(re(x)) - re (x)*--(re(x)) + 2*--(im(x))*im(x)*re(x)|  + 3*|re (x)*--(im(x)) - im (x)*--(im(x)) + 2*--(re(x))*im(x)*re(x)|  + \- im (x) + 3*re (x)/*|re (x)*---(im(x)) - im (x)*---(im(x)) - 2*|--(im(x))| *im(x) + 2*|--(re(x))| *im(x) + 2*---(re(x))*im(x)*re(x) + 4*--(im(x))*--(re(x))*re(x)|*im(x) - \re (x) - 3*im (x)/*|im (x)*---(re(x)) - re (x)*---(re(x)) - 2*|--(re(x))| *re(x) + 2*|--(im(x))| *re(x) + 2*---(im(x))*im(x)*re(x) + 4*--(im(x))*--(re(x))*im(x)|*re(x)||
  |                                                                  /         2                   2                       2                      2             2                                                 \                                                                     /         2                   2                       2                      2             2                                                 \   27*|\- im (x) + 3*re (x)/*|re (x)*--(im(x)) - im (x)*--(im(x)) + 2*--(re(x))*im(x)*re(x)|*im(x) - \re (x) - 3*im (x)/*|im (x)*--(re(x)) - re (x)*--(re(x)) + 2*--(im(x))*im(x)*re(x)|*re(x)|                          /               3            3                   3                       2                           2                    3                                    2                              2                              2             \                             /               3            3                   3                       2                           2                    3                                    2                              2                              2             \           \                      \       dx                 dx            dx                   /                             \       dx                 dx            dx                   /      / |  \       dx                 dx            dx                   /      \       dx                 dx            dx                   /                          |         2                   2            \dx       /            \dx       /              2                        dx        dx             |                             |         2                   2            \dx       /            \dx       /              2                        dx        dx             |      ||
  |  /  2    d             2    d             d                    \ |  2     d             2     d             /d        \            /d        \             d                         d         d              |     /  2    d             2    d             d                    \ |  2     d             2     d             /d        \            /d        \             d                         d         d              |      \                      \       dx                 dx            dx                   /                             \       dx                 dx            dx                   /      /    /    2          2   \ |    /d        \      2     d             2     d             /d        \  d             d          d                    d                         d          d                   d          d                   d          d              |         /  2          2   \ |    /d        \      2     d             2     d             /d        \  d             d          d                    d                         d          d                   d          d                   d          d              |                                                                                                                                                                                                     \                                                                                                                                                                \       dx                  dx                                                           dx                                                  /                             \       dx                  dx                                                           dx                                                  /      /|
3*|9*|im (x)*--(re(x)) - re (x)*--(re(x)) + 2*--(im(x))*im(x)*re(x)|*|im (x)*---(re(x)) - re (x)*---(re(x)) - 2*|--(re(x))| *re(x) + 2*|--(im(x))| *re(x) + 2*---(im(x))*im(x)*re(x) + 4*--(im(x))*--(re(x))*im(x)| + 9*|re (x)*--(im(x)) - im (x)*--(im(x)) + 2*--(re(x))*im(x)*re(x)|*|re (x)*---(im(x)) - im (x)*---(im(x)) - 2*|--(im(x))| *im(x) + 2*|--(re(x))| *im(x) + 2*---(re(x))*im(x)*re(x) + 4*--(im(x))*--(re(x))*re(x)| + --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + \- im (x) + 3*re (x)/*|- 2*|--(im(x))|  + re (x)*---(im(x)) - im (x)*---(im(x)) + 6*|--(re(x))| *--(im(x)) - 6*--(im(x))*---(im(x))*im(x) + 2*---(re(x))*im(x)*re(x) + 6*--(im(x))*---(re(x))*re(x) + 6*--(re(x))*---(im(x))*re(x) + 6*--(re(x))*---(re(x))*im(x)|*im(x) - \re (x) - 3*im (x)/*|- 2*|--(re(x))|  + im (x)*---(re(x)) - re (x)*---(re(x)) + 6*|--(im(x))| *--(re(x)) - 6*--(re(x))*---(re(x))*re(x) + 2*---(im(x))*im(x)*re(x) + 6*--(im(x))*---(im(x))*re(x) + 6*--(im(x))*---(re(x))*im(x) + 6*--(re(x))*---(im(x))*im(x)|*re(x) - ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|
  |  \       dx                 dx            dx                   / |         2                   2            \dx       /            \dx       /              2                        dx        dx             |     \       dx                 dx            dx                   / |         2                   2            \dx       /            \dx       /              2                        dx        dx             |                                                                                                   4                                                                                                                     |    \dx       /             3                   3            \dx       /  dx            dx          2                    3                        dx          2                  dx          2                  dx          2             |                             |    \dx       /             3                   3            \dx       /  dx            dx          2                    3                        dx          2                  dx          2                  dx          2             |                                                                                                                                                                                                                                                                                                                                                         2                                                                                                                                                                                                                                                                                                                                            |
  |                                                                  \       dx                  dx                                                           dx                                                  /                                                                     \       dx                  dx                                                           dx                                                  /                                                                                               | 3|                                                                                                                      \                          dx                  dx                                                  dx                   dx                                   dx                             dx                             dx              /                             \                          dx                  dx                                                  dx                   dx                                   dx                             dx                             dx              /                                                                                                                                                                                                                                                                                                                                                     | 3|                                                                                                                                                                                                                                                                                                                                             |
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                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |x |
$$\frac{1}{\left|{x^{3}}\right|} \left(\frac{81}{\left|{x^{3}}\right|^{4}} \left(- \left(\left(\Re{x}\right)^{2} - 3 \left(\Im{x}\right)^{2}\right) \left(- \left(\Re{x}\right)^{2} \frac{d}{d x} \Re{x} + 2 \Re{x} \Im{x} \frac{d}{d x} \Im{x} + \left(\Im{x}\right)^{2} \frac{d}{d x} \Re{x}\right) \Re{x} + \left(3 \left(\Re{x}\right)^{2} - \left(\Im{x}\right)^{2}\right) \left(\left(\Re{x}\right)^{2} \frac{d}{d x} \Im{x} + 2 \Re{x} \Im{x} \frac{d}{d x} \Re{x} - \left(\Im{x}\right)^{2} \frac{d}{d x} \Im{x}\right) \Im{x}\right)^{3} - \frac{27}{\left|{x^{3}}\right|^{2}} \left(- \left(\left(\Re{x}\right)^{2} - 3 \left(\Im{x}\right)^{2}\right) \left(- \left(\Re{x}\right)^{2} \frac{d}{d x} \Re{x} + 2 \Re{x} \Im{x} \frac{d}{d x} \Im{x} + \left(\Im{x}\right)^{2} \frac{d}{d x} \Re{x}\right) \Re{x} + \left(3 \left(\Re{x}\right)^{2} - \left(\Im{x}\right)^{2}\right) \left(\left(\Re{x}\right)^{2} \frac{d}{d x} \Im{x} + 2 \Re{x} \Im{x} \frac{d}{d x} \Re{x} - \left(\Im{x}\right)^{2} \frac{d}{d x} \Im{x}\right) \Im{x}\right) \left(- \left(\left(\Re{x}\right)^{2} - 3 \left(\Im{x}\right)^{2}\right) \left(- \left(\Re{x}\right)^{2} \frac{d^{2}}{d x^{2}} \Re{x} + 2 \Re{x} \Im{x} \frac{d^{2}}{d x^{2}} \Im{x} - 2 \Re{x} \frac{d}{d x} \Re{x}^{2} + 2 \Re{x} \frac{d}{d x} \Im{x}^{2} + \left(\Im{x}\right)^{2} \frac{d^{2}}{d x^{2}} \Re{x} + 4 \Im{x} \frac{d}{d x} \Re{x} \frac{d}{d x} \Im{x}\right) \Re{x} + \left(3 \left(\Re{x}\right)^{2} - \left(\Im{x}\right)^{2}\right) \left(\left(\Re{x}\right)^{2} \frac{d^{2}}{d x^{2}} \Im{x} + 2 \Re{x} \Im{x} \frac{d^{2}}{d x^{2}} \Re{x} + 4 \Re{x} \frac{d}{d x} \Re{x} \frac{d}{d x} \Im{x} - \left(\Im{x}\right)^{2} \frac{d^{2}}{d x^{2}} \Im{x} + 2 \Im{x} \frac{d}{d x} \Re{x}^{2} - 2 \Im{x} \frac{d}{d x} \Im{x}^{2}\right) \Im{x} + 3 \left(- \left(\Re{x}\right)^{2} \frac{d}{d x} \Re{x} + 2 \Re{x} \Im{x} \frac{d}{d x} \Im{x} + \left(\Im{x}\right)^{2} \frac{d}{d x} \Re{x}\right)^{2} + 3 \left(\left(\Re{x}\right)^{2} \frac{d}{d x} \Im{x} + 2 \Re{x} \Im{x} \frac{d}{d x} \Re{x} - \left(\Im{x}\right)^{2} \frac{d}{d x} \Im{x}\right)^{2}\right) - 3 \left(\left(\Re{x}\right)^{2} - 3 \left(\Im{x}\right)^{2}\right) \left(- \left(\Re{x}\right)^{2} \frac{d^{3}}{d x^{3}} \Re{x} + 2 \Re{x} \Im{x} \frac{d^{3}}{d x^{3}} \Im{x} - 6 \Re{x} \frac{d}{d x} \Re{x} \frac{d^{2}}{d x^{2}} \Re{x} + 6 \Re{x} \frac{d}{d x} \Im{x} \frac{d^{2}}{d x^{2}} \Im{x} + \left(\Im{x}\right)^{2} \frac{d^{3}}{d x^{3}} \Re{x} + 6 \Im{x} \frac{d}{d x} \Re{x} \frac{d^{2}}{d x^{2}} \Im{x} + 6 \Im{x} \frac{d}{d x} \Im{x} \frac{d^{2}}{d x^{2}} \Re{x} - 2 \frac{d}{d x} \Re{x}^{3} + 6 \frac{d}{d x} \Re{x} \frac{d}{d x} \Im{x}^{2}\right) \Re{x} + 3 \left(3 \left(\Re{x}\right)^{2} - \left(\Im{x}\right)^{2}\right) \left(\left(\Re{x}\right)^{2} \frac{d^{3}}{d x^{3}} \Im{x} + 2 \Re{x} \Im{x} \frac{d^{3}}{d x^{3}} \Re{x} + 6 \Re{x} \frac{d}{d x} \Re{x} \frac{d^{2}}{d x^{2}} \Im{x} + 6 \Re{x} \frac{d}{d x} \Im{x} \frac{d^{2}}{d x^{2}} \Re{x} - \left(\Im{x}\right)^{2} \frac{d^{3}}{d x^{3}} \Im{x} + 6 \Im{x} \frac{d}{d x} \Re{x} \frac{d^{2}}{d x^{2}} \Re{x} - 6 \Im{x} \frac{d}{d x} \Im{x} \frac{d^{2}}{d x^{2}} \Im{x} + 6 \frac{d}{d x} \Re{x}^{2} \frac{d}{d x} \Im{x} - 2 \frac{d}{d x} \Im{x}^{3}\right) \Im{x} + 27 \left(- \left(\Re{x}\right)^{2} \frac{d}{d x} \Re{x} + 2 \Re{x} \Im{x} \frac{d}{d x} \Im{x} + \left(\Im{x}\right)^{2} \frac{d}{d x} \Re{x}\right) \left(- \left(\Re{x}\right)^{2} \frac{d^{2}}{d x^{2}} \Re{x} + 2 \Re{x} \Im{x} \frac{d^{2}}{d x^{2}} \Im{x} - 2 \Re{x} \frac{d}{d x} \Re{x}^{2} + 2 \Re{x} \frac{d}{d x} \Im{x}^{2} + \left(\Im{x}\right)^{2} \frac{d^{2}}{d x^{2}} \Re{x} + 4 \Im{x} \frac{d}{d x} \Re{x} \frac{d}{d x} \Im{x}\right) + 27 \left(\left(\Re{x}\right)^{2} \frac{d}{d x} \Im{x} + 2 \Re{x} \Im{x} \frac{d}{d x} \Re{x} - \left(\Im{x}\right)^{2} \frac{d}{d x} \Im{x}\right) \left(\left(\Re{x}\right)^{2} \frac{d^{2}}{d x^{2}} \Im{x} + 2 \Re{x} \Im{x} \frac{d^{2}}{d x^{2}} \Re{x} + 4 \Re{x} \frac{d}{d x} \Re{x} \frac{d}{d x} \Im{x} - \left(\Im{x}\right)^{2} \frac{d^{2}}{d x^{2}} \Im{x} + 2 \Im{x} \frac{d}{d x} \Re{x}^{2} - 2 \Im{x} \frac{d}{d x} \Im{x}^{2}\right)\right)$$
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