Производная acos(x)^(1/x)

Вы ввели

x _________
\/ acos(x) 

$$\operatorname{acos}^{\frac{1}{x}}{\left (x \right )}$$

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

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

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Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

             /                                    3                                                                                                                                                                                                                                                                                                 \
             |/log(acos(x))            1         \                                                                                                                    /log(acos(x))            1         \ /           1            2*log(acos(x))            1                       2           \                                                 |
             ||------------ + -------------------|                                                                                                                  3*|------------ + -------------------|*|- ------------------- + -------------- + -------------------- + ----------------------|                                                 |
             ||     x            ________        |                                                                                                                    |     x            ________        | |          3/2                  3           /      2\     2            ________        |                                                 |
             ||                 /      2         |                                                                                                                    |                 /      2         | |  /     2\                    x          x*\-1 + x /*acos (x)    2   /      2         |                                                 |
 x _________ |\               \/  1 - x  *acos(x)/             3            6*log(acos(x))             2                       2                      3*x             \               \/  1 - x  *acos(x)/ \  \1 - x /   *acos(x)                                           x *\/  1 - x  *acos(x)/             3                       6           |
-\/ acos(x) *|------------------------------------- + ------------------- + -------------- - --------------------- + ---------------------- + ------------------- + ------------------------------------------------------------------------------------------------------------------------------- + --------------------- + ----------------------|
             |                   3                             2                   4                   3/2                     3/2                    5/2                                                                          x                                                                   2 /      2\     2            ________        |
             |                  x                     /      2\      2            x            /     2\                /     2\        3      /     2\                                                                                                                                                x *\-1 + x /*acos (x)    3   /      2         |
             \                                        \-1 + x / *acos (x)                    x*\1 - x /   *acos(x)   x*\1 - x /   *acos (x)   \1 - x /   *acos(x)                                                                                                                                                             x *\/  1 - x  *acos(x)/

$$- \left(\frac{3 x}{\left(- x^{2} + 1\right)^{\frac{5}{2}} \operatorname{acos}{\left (x \right )}} + \frac{3}{\left(x^{2} - 1\right)^{2} \operatorname{acos}^{2}{\left (x \right )}} + \frac{3}{x} \left(\frac{1}{\sqrt{- x^{2} + 1} \operatorname{acos}{\left (x \right )}} + \frac{1}{x} \log{\left (\operatorname{acos}{\left (x \right )} \right )}\right) \left(- \frac{1}{\left(- x^{2} + 1\right)^{\frac{3}{2}} \operatorname{acos}{\left (x \right )}} + \frac{1}{x \left(x^{2} - 1\right) \operatorname{acos}^{2}{\left (x \right )}} + \frac{2}{x^{2} \sqrt{- x^{2} + 1} \operatorname{acos}{\left (x \right )}} + \frac{2}{x^{3}} \log{\left (\operatorname{acos}{\left (x \right )} \right )}\right) - \frac{2}{x \left(- x^{2} + 1\right)^{\frac{3}{2}} \operatorname{acos}{\left (x \right )}} + \frac{2}{x \left(- x^{2} + 1\right)^{\frac{3}{2}} \operatorname{acos}^{3}{\left (x \right )}} + \frac{3}{x^{2} \left(x^{2} - 1\right) \operatorname{acos}^{2}{\left (x \right )}} + \frac{1}{x^{3}} \left(\frac{1}{\sqrt{- x^{2} + 1} \operatorname{acos}{\left (x \right )}} + \frac{1}{x} \log{\left (\operatorname{acos}{\left (x \right )} \right )}\right)^{3} + \frac{6}{x^{3} \sqrt{- x^{2} + 1} \operatorname{acos}{\left (x \right )}} + \frac{6}{x^{4}} \log{\left (\operatorname{acos}{\left (x \right )} \right )}\right) \operatorname{acos}^{\frac{1}{x}}{\left (x \right )}$$