Производная acos(x)^(3*x)

Вы ввели

    3*x   
acos   (x)

$$\operatorname{acos}^{3 x}{\left (x \right )}$$

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

   $\left(3 x\right)^{3 x} \left(\log{\left (3 x \right )} + 1\right)$

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Ответ:

$\left(3 x\right)^{3 x} \left(\log{\left (3 x \right )} + 1\right)$

Первая производная

    3*x    /                         3*x        \
acos   (x)*|3*log(acos(x)) - -------------------|
           |                    ________        |
           |                   /      2         |
           \                 \/  1 - x  *acos(x)/

$$\left(- \frac{3 x}{\sqrt{- x^{2} + 1} \operatorname{acos}{\left (x \right )}} + 3 \log{\left (\operatorname{acos}{\left (x \right )} \right )}\right) \operatorname{acos}^{3 x}{\left (x \right )}$$

Вторая производная

             /                                                               2                        \
             |                                                2             x                x        |
             |                                           ----------- + ----------- - -----------------|
             |                                              ________           3/2   /      2\        |
             |                                       2     /      2    /     2\      \-1 + x /*acos(x)|
      3*x    |  /                         x         \    \/  1 - x     \1 - x /                       |
3*acos   (x)*|3*|-log(acos(x)) + -------------------|  - ---------------------------------------------|
             |  |                   ________        |                       acos(x)                   |
             |  |                  /      2         |                                                 |
             \  \                \/  1 - x  *acos(x)/                                                 /

$$3 \left(3 \left(\frac{x}{\sqrt{- x^{2} + 1} \operatorname{acos}{\left (x \right )}} - \log{\left (\operatorname{acos}{\left (x \right )} \right )}\right)^{2} - \frac{1}{\operatorname{acos}{\left (x \right )}} \left(\frac{x^{2}}{\left(- x^{2} + 1\right)^{\frac{3}{2}}} - \frac{x}{\left(x^{2} - 1\right) \operatorname{acos}{\left (x \right )}} + \frac{2}{\sqrt{- x^{2} + 1}}\right)\right) \operatorname{acos}^{3 x}{\left (x \right )}$$

Третья производная

             /                                                                          3                                                     2                                                  /                    2                        \\
             |                                                       3               3*x           4*x               2*x                   3*x             /                         x         \ |     2             x                x        ||
             |                                             - ----------------- + ----------- + ----------- + -------------------- + ------------------   9*|-log(acos(x)) + -------------------|*|----------- + ----------- - -----------------||
             |                                               /      2\                   5/2           3/2           3/2                     2             |                   ________        | |   ________           3/2   /      2\        ||
             |                                         3     \-1 + x /*acos(x)   /     2\      /     2\      /     2\        2      /      2\              |                  /      2         | |  /      2    /     2\      \-1 + x /*acos(x)||
      3*x    |    /                         x         \                          \1 - x /      \1 - x /      \1 - x /   *acos (x)   \-1 + x / *acos(x)     \                \/  1 - x  *acos(x)/ \\/  1 - x     \1 - x /                       /|
3*acos   (x)*|- 9*|-log(acos(x)) + -------------------|  - ------------------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------|
             |    |                   ________        |                                              acos(x)                                                                                     acos(x)                                        |
             |    |                  /      2         |                                                                                                                                                                                         |
             \    \                \/  1 - x  *acos(x)/                                                                                                                                                                                         /

$$3 \left(- 9 \left(\frac{x}{\sqrt{- x^{2} + 1} \operatorname{acos}{\left (x \right )}} - \log{\left (\operatorname{acos}{\left (x \right )} \right )}\right)^{3} + \frac{9}{\operatorname{acos}{\left (x \right )}} \left(\frac{x}{\sqrt{- x^{2} + 1} \operatorname{acos}{\left (x \right )}} - \log{\left (\operatorname{acos}{\left (x \right )} \right )}\right) \left(\frac{x^{2}}{\left(- x^{2} + 1\right)^{\frac{3}{2}}} - \frac{x}{\left(x^{2} - 1\right) \operatorname{acos}{\left (x \right )}} + \frac{2}{\sqrt{- x^{2} + 1}}\right) - \frac{1}{\operatorname{acos}{\left (x \right )}} \left(\frac{3 x^{3}}{\left(- x^{2} + 1\right)^{\frac{5}{2}}} + \frac{3 x^{2}}{\left(x^{2} - 1\right)^{2} \operatorname{acos}{\left (x \right )}} + \frac{4 x}{\left(- x^{2} + 1\right)^{\frac{3}{2}}} + \frac{2 x}{\left(- x^{2} + 1\right)^{\frac{3}{2}} \operatorname{acos}^{2}{\left (x \right )}} - \frac{3}{\left(x^{2} - 1\right) \operatorname{acos}{\left (x \right )}}\right)\right) \operatorname{acos}^{3 x}{\left (x \right )}$$