Производная (log(x+e))^acos(x)

Вы ввели

   acos(x)       
log       (x + E)

$$\log^{\operatorname{acos}{\left (x \right )}}{\left (x + e \right )}$$

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

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

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Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

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

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