Подробное решение
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Не могу найти шаги в поиске этой производной.
Но производная
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Теперь упростим:
Ответ:
/ ________ \
| / 2 / 1 / 1\\|
| \/ 1 + x *|- --------------- + asin|1*-|||
| | ________ \ x/||
________ | | / 1 ||
/ 2 | / / 1\\ | x* / 1 - -- ||
\/ 1 + x |x*log|x*asin|1*-|| | / 2 ||
/ / 1\\ | \ \ x// \ \/ x /|
|x*asin|1*-|| *|------------------ + -------------------------------------------|
\ \ x// | ________ / 1\ |
| / 2 x*asin|1*-| |
\ \/ 1 + x \ x/ /
$$\left(x \operatorname{asin}{\left(1 \cdot \frac{1}{x} \right)}\right)^{\sqrt{x^{2} + 1}} \left(\frac{x \log{\left(x \operatorname{asin}{\left(1 \cdot \frac{1}{x} \right)} \right)}}{\sqrt{x^{2} + 1}} + \frac{\sqrt{x^{2} + 1} \left(\operatorname{asin}{\left(1 \cdot \frac{1}{x} \right)} - \frac{1}{x \sqrt{1 - \frac{1}{x^{2}}}}\right)}{x \operatorname{asin}{\left(1 \cdot \frac{1}{x} \right)}}\right)$$
/ 2 \
|/ ________ \ ________ ________ |
|| / 2 / 1 /1\\| / 1 /1\\ / 2 / 1 /1\\ / 2 / 1 /1\\|
|| \/ 1 + x *|- --------------- + asin|-||| 2*|- --------------- + asin|-|| \/ 1 + x *|- --------------- + asin|-|| \/ 1 + x *|- --------------- + asin|-|||
|| | ________ \x/|| | ________ \x/| | ________ \x/| | ________ \x/||
________ || | / 1 || | / 1 | | / 1 | | / 1 ||
/ 2 || / /1\\ | x* / 1 - -- || / /1\\ 2 / /1\\ | x* / 1 - -- | ________ | x* / 1 - -- | | x* / 1 - -- ||
\/ 1 + x ||x*log|x*asin|-|| | / 2 || log|x*asin|-|| x *log|x*asin|-|| | / 2 | / 2 | / 2 | | / 2 ||
/ /1\\ || \ \x// \ \/ x /| \ \x// \ \x// \ \/ x / \/ 1 + x \ \/ x / \ \/ x /|
|x*asin|-|| *||---------------- + -----------------------------------------| + -------------- - ----------------- + ------------------------------- + ---------------------- - ----------------------------------------- + -----------------------------------------|
\ \x// || ________ /1\ | ________ 3/2 ________ 3/2 2 /1\ ________ |
|| / 2 x*asin|-| | / 2 / 2\ / 2 /1\ 5 / 1 \ /1\ x *asin|-| 3 / 1 2/1\ |
|\ \/ 1 + x \x/ / \/ 1 + x \1 + x / \/ 1 + x *asin|-| x *|1 - --| *asin|-| \x/ x * / 1 - -- *asin |-| |
| \x/ | 2| \x/ / 2 \x/ |
\ \ x / \/ x /
$$\left(x \operatorname{asin}{\left(\frac{1}{x} \right)}\right)^{\sqrt{x^{2} + 1}} \left(\left(\frac{x \log{\left(x \operatorname{asin}{\left(\frac{1}{x} \right)} \right)}}{\sqrt{x^{2} + 1}} + \frac{\sqrt{x^{2} + 1} \left(\operatorname{asin}{\left(\frac{1}{x} \right)} - \frac{1}{x \sqrt{1 - \frac{1}{x^{2}}}}\right)}{x \operatorname{asin}{\left(\frac{1}{x} \right)}}\right)^{2} - \frac{x^{2} \log{\left(x \operatorname{asin}{\left(\frac{1}{x} \right)} \right)}}{\left(x^{2} + 1\right)^{\frac{3}{2}}} + \frac{\log{\left(x \operatorname{asin}{\left(\frac{1}{x} \right)} \right)}}{\sqrt{x^{2} + 1}} + \frac{2 \left(\operatorname{asin}{\left(\frac{1}{x} \right)} - \frac{1}{x \sqrt{1 - \frac{1}{x^{2}}}}\right)}{\sqrt{x^{2} + 1} \operatorname{asin}{\left(\frac{1}{x} \right)}} - \frac{\sqrt{x^{2} + 1} \left(\operatorname{asin}{\left(\frac{1}{x} \right)} - \frac{1}{x \sqrt{1 - \frac{1}{x^{2}}}}\right)}{x^{2} \operatorname{asin}{\left(\frac{1}{x} \right)}} + \frac{\sqrt{x^{2} + 1} \left(\operatorname{asin}{\left(\frac{1}{x} \right)} - \frac{1}{x \sqrt{1 - \frac{1}{x^{2}}}}\right)}{x^{3} \sqrt{1 - \frac{1}{x^{2}}} \operatorname{asin}^{2}{\left(\frac{1}{x} \right)}} + \frac{\sqrt{x^{2} + 1}}{x^{5} \left(1 - \frac{1}{x^{2}}\right)^{\frac{3}{2}} \operatorname{asin}{\left(\frac{1}{x} \right)}}\right)$$
/ 3 \
|/ ________ \ / ________ \ / ________ ________ \ ________ ________ ________ ________ |
|| / 2 / 1 /1\\| | / 2 / 1 /1\\| | / 1 /1\\ / 2 / 1 /1\\ / 2 / 1 /1\\| / 1 /1\\ / 2 / 1 /1\\ / 2 / 1 /1\\ / 2 / 1 /1\\ / 2 / 1 /1\\ / 1 /1\\ |
|| \/ 1 + x *|- --------------- + asin|-||| | \/ 1 + x *|- --------------- + asin|-||| | 2*|- --------------- + asin|-|| \/ 1 + x *|- --------------- + asin|-|| \/ 1 + x *|- --------------- + asin|-||| 3*x*|- --------------- + asin|-|| 2*\/ 1 + x *|- --------------- + asin|-|| \/ 1 + x *|- --------------- + asin|-|| 4*\/ 1 + x *|- --------------- + asin|-|| 2*\/ 1 + x *|- --------------- + asin|-|| 3*|- --------------- + asin|-|| |
|| | ________ \x/|| | | ________ \x/|| | | ________ \x/| | ________ \x/| | ________ \x/|| | ________ \x/| | ________ \x/| | ________ \x/| | ________ \x/| | ________ \x/| | ________ \x/| |
________ || | / 1 || | | / 1 || | | / 1 | | / 1 | | / 1 || | / 1 | | / 1 | | / 1 | | / 1 | | / 1 | | / 1 | |
/ 2 || / /1\\ | x* / 1 - -- || | / /1\\ | x* / 1 - -- || | / /1\\ 2 / /1\\ | x* / 1 - -- | ________ | x* / 1 - -- | | x* / 1 - -- || / /1\\ 3 / /1\\ ________ | x* / 1 - -- | ________ ________ | x* / 1 - -- | | x* / 1 - -- | | x* / 1 - -- | | x* / 1 - -- | | x* / 1 - -- | |
\/ 1 + x ||x*log|x*asin|-|| | / 2 || |x*log|x*asin|-|| | / 2 || |log|x*asin|-|| x *log|x*asin|-|| | / 2 | / 2 | / 2 | | / 2 || 3*x*log|x*asin|-|| 3*x *log|x*asin|-|| / 2 | / 2 | / 2 / 2 | / 2 | | / 2 | | / 2 | | / 2 | | / 2 | |
/ /1\\ || \ \x// \ \/ x /| | \ \x// \ \/ x /| | \ \x// \ \x// \ \/ x / \/ 1 + x \ \/ x / \ \/ x /| \ \x// \ \x// 6*\/ 1 + x \ \/ x / 3*\/ 1 + x 2*\/ 1 + x \ \/ x / 3 \ \/ x / \ \/ x / \ \/ x / \ \/ x / |
|x*asin|-|| *||---------------- + -----------------------------------------| + 3*|---------------- + -----------------------------------------|*|-------------- - ----------------- + ------------------------------- + ---------------------- - ----------------------------------------- + -----------------------------------------| - ------------------ + ------------------- - ---------------------- - --------------------------------- - ---------------------- + --------------------- + ------------------------------------------- + ---------------------------------- - ----------------------------------------- - ------------------------------------------- + ------------------------------------------- + -------------------------------------|
\ \x// || ________ /1\ | | ________ /1\ | | ________ 3/2 ________ 3/2 2 /1\ ________ | 3/2 5/2 3/2 3/2 5/2 2 3 /1\ ________ 3/2 3/2 ________ 5 / 1 \ 3/1\ ________ ________ |
|| / 2 x*asin|-| | | / 2 x*asin|-| | | / 2 / 2\ / 2 /1\ 5 / 1 \ /1\ x *asin|-| 3 / 1 2/1\ | / 2\ / 2\ 6 / 1 \ /1\ / 2\ /1\ 8 / 1 \ /1\ 7 / 1 \ 2/1\ x *asin|-| 4 / 2 / 1 \ /1\ 6 / 1 \ 2/1\ 4 / 1 2/1\ x *|1 - --|*asin |-| 2 / 2 / 1 2/1\|
|\ \/ 1 + x \x/ / \ \/ 1 + x \x/ / | \/ 1 + x \1 + x / \/ 1 + x *asin|-| x *|1 - --| *asin|-| \x/ x * / 1 - -- *asin |-| | \1 + x / \1 + x / x *|1 - --| *asin|-| \1 + x / *asin|-| x *|1 - --| *asin|-| x *|1 - --| *asin |-| \x/ x *\/ 1 + x *|1 - --| *asin|-| x *|1 - --| *asin |-| x * / 1 - -- *asin |-| | 2| \x/ x *\/ 1 + x * / 1 - -- *asin |-||
| | \x/ | 2| \x/ / 2 \x/ | | 2| \x/ \x/ | 2| \x/ | 2| \x/ | 2| \x/ | 2| \x/ / 2 \x/ \ x / / 2 \x/|
\ \ \ x / \/ x / \ x / \ x / \ x / \ x / \ x / \/ x \/ x /
$$\left(x \operatorname{asin}{\left(\frac{1}{x} \right)}\right)^{\sqrt{x^{2} + 1}} \cdot \left(\left(\frac{x \log{\left(x \operatorname{asin}{\left(\frac{1}{x} \right)} \right)}}{\sqrt{x^{2} + 1}} + \frac{\sqrt{x^{2} + 1} \left(\operatorname{asin}{\left(\frac{1}{x} \right)} - \frac{1}{x \sqrt{1 - \frac{1}{x^{2}}}}\right)}{x \operatorname{asin}{\left(\frac{1}{x} \right)}}\right)^{3} + 3 \left(\frac{x \log{\left(x \operatorname{asin}{\left(\frac{1}{x} \right)} \right)}}{\sqrt{x^{2} + 1}} + \frac{\sqrt{x^{2} + 1} \left(\operatorname{asin}{\left(\frac{1}{x} \right)} - \frac{1}{x \sqrt{1 - \frac{1}{x^{2}}}}\right)}{x \operatorname{asin}{\left(\frac{1}{x} \right)}}\right) \left(- \frac{x^{2} \log{\left(x \operatorname{asin}{\left(\frac{1}{x} \right)} \right)}}{\left(x^{2} + 1\right)^{\frac{3}{2}}} + \frac{\log{\left(x \operatorname{asin}{\left(\frac{1}{x} \right)} \right)}}{\sqrt{x^{2} + 1}} + \frac{2 \left(\operatorname{asin}{\left(\frac{1}{x} \right)} - \frac{1}{x \sqrt{1 - \frac{1}{x^{2}}}}\right)}{\sqrt{x^{2} + 1} \operatorname{asin}{\left(\frac{1}{x} \right)}} - \frac{\sqrt{x^{2} + 1} \left(\operatorname{asin}{\left(\frac{1}{x} \right)} - \frac{1}{x \sqrt{1 - \frac{1}{x^{2}}}}\right)}{x^{2} \operatorname{asin}{\left(\frac{1}{x} \right)}} + \frac{\sqrt{x^{2} + 1} \left(\operatorname{asin}{\left(\frac{1}{x} \right)} - \frac{1}{x \sqrt{1 - \frac{1}{x^{2}}}}\right)}{x^{3} \sqrt{1 - \frac{1}{x^{2}}} \operatorname{asin}^{2}{\left(\frac{1}{x} \right)}} + \frac{\sqrt{x^{2} + 1}}{x^{5} \left(1 - \frac{1}{x^{2}}\right)^{\frac{3}{2}} \operatorname{asin}{\left(\frac{1}{x} \right)}}\right) + \frac{3 x^{3} \log{\left(x \operatorname{asin}{\left(\frac{1}{x} \right)} \right)}}{\left(x^{2} + 1\right)^{\frac{5}{2}}} - \frac{3 x \log{\left(x \operatorname{asin}{\left(\frac{1}{x} \right)} \right)}}{\left(x^{2} + 1\right)^{\frac{3}{2}}} - \frac{3 x \left(\operatorname{asin}{\left(\frac{1}{x} \right)} - \frac{1}{x \sqrt{1 - \frac{1}{x^{2}}}}\right)}{\left(x^{2} + 1\right)^{\frac{3}{2}} \operatorname{asin}{\left(\frac{1}{x} \right)}} + \frac{2 \sqrt{x^{2} + 1} \left(\operatorname{asin}{\left(\frac{1}{x} \right)} - \frac{1}{x \sqrt{1 - \frac{1}{x^{2}}}}\right)}{x^{3} \operatorname{asin}{\left(\frac{1}{x} \right)}} + \frac{3 \left(\operatorname{asin}{\left(\frac{1}{x} \right)} - \frac{1}{x \sqrt{1 - \frac{1}{x^{2}}}}\right)}{x^{2} \sqrt{1 - \frac{1}{x^{2}}} \sqrt{x^{2} + 1} \operatorname{asin}^{2}{\left(\frac{1}{x} \right)}} - \frac{4 \sqrt{x^{2} + 1} \left(\operatorname{asin}{\left(\frac{1}{x} \right)} - \frac{1}{x \sqrt{1 - \frac{1}{x^{2}}}}\right)}{x^{4} \sqrt{1 - \frac{1}{x^{2}}} \operatorname{asin}^{2}{\left(\frac{1}{x} \right)}} + \frac{2 \sqrt{x^{2} + 1} \left(\operatorname{asin}{\left(\frac{1}{x} \right)} - \frac{1}{x \sqrt{1 - \frac{1}{x^{2}}}}\right)}{x^{5} \cdot \left(1 - \frac{1}{x^{2}}\right) \operatorname{asin}^{3}{\left(\frac{1}{x} \right)}} + \frac{3}{x^{4} \left(1 - \frac{1}{x^{2}}\right)^{\frac{3}{2}} \sqrt{x^{2} + 1} \operatorname{asin}{\left(\frac{1}{x} \right)}} - \frac{\sqrt{x^{2} + 1} \left(\operatorname{asin}{\left(\frac{1}{x} \right)} - \frac{1}{x \sqrt{1 - \frac{1}{x^{2}}}}\right)}{x^{6} \left(1 - \frac{1}{x^{2}}\right)^{\frac{3}{2}} \operatorname{asin}^{2}{\left(\frac{1}{x} \right)}} - \frac{6 \sqrt{x^{2} + 1}}{x^{6} \left(1 - \frac{1}{x^{2}}\right)^{\frac{3}{2}} \operatorname{asin}{\left(\frac{1}{x} \right)}} + \frac{2 \sqrt{x^{2} + 1}}{x^{7} \left(1 - \frac{1}{x^{2}}\right)^{2} \operatorname{asin}^{2}{\left(\frac{1}{x} \right)}} - \frac{3 \sqrt{x^{2} + 1}}{x^{8} \left(1 - \frac{1}{x^{2}}\right)^{\frac{5}{2}} \operatorname{asin}{\left(\frac{1}{x} \right)}}\right)$$