Производная (|x|)^(x+a)

Вы ввели

   x + a
|x|     

$$\left|{x}\right|^{a + x}$$

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

   $\left(a + x\right)^{a + x} \left(\log{\left (a + x \right )} + 1\right)$

2. Теперь упростим:

   $\left(a + x\right)^{a + x} \left(\log{\left (a + x \right )} + 1\right)$

---

Ответ:

$\left(a + x\right)^{a + x} \left(\log{\left (a + x \right )} + 1\right)$

Первая производная

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

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

Вторая производная

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

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

Третья производная

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

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