Господин Экзамен
Производная (cos(x)^(2))^(1/x)
Решение
Вы ввели
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_________ x / 2 \/ cos (x)
$$\left(\cos^{2}{\left(x \right)}\right)^{1 \cdot \frac{1}{x}}$$
/ _________\ d |x / 2 | --\\/ cos (x) / dx
$$\frac{d}{d x} \left(\cos^{2}{\left(x \right)}\right)^{1 \cdot \frac{1}{x}}$$
Подробное решение
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Не могу найти шаги в поиске этой производной.
Но производная
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Теперь упростим:
Ответ:
Первая производная
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2
- / / 2 \ \
x | log\cos (x)/ 2*sin(x)|
|cos(x)| *|- ------------ - --------|
| 2 x*cos(x)|
\ x /
$$\left(- \frac{2 \sin{\left(x \right)}}{x \cos{\left(x \right)}} - \frac{\log{\left(\cos^{2}{\left(x \right)} \right)}}{x^{2}}\right) \left|{\cos{\left(x \right)}}\right|^{\frac{2}{x}}$$
Вторая производная
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/ / / 2 \ \ \
2 | |log\cos (x)/ 2*sin(x)| /log(|cos(x)|) sign(cos(x))*sin(x)\ |
- | / 2 \ 2 |------------ + --------|*|------------- + -------------------| |
x | log\cos (x)/ sin (x) \ x cos(x) / \ x |cos(x)| / 2*sin(x)|
2*|cos(x)| *|-1 + ------------ - ------- + --------------------------------------------------------------- + --------|
| 2 2 x x*cos(x)|
\ x cos (x) /
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x
$$\frac{2 \left(\frac{\left(\frac{2 \sin{\left(x \right)}}{\cos{\left(x \right)}} + \frac{\log{\left(\cos^{2}{\left(x \right)} \right)}}{x}\right) \left(\frac{\sin{\left(x \right)} \operatorname{sign}{\left(\cos{\left(x \right)} \right)}}{\left|{\cos{\left(x \right)}}\right|} + \frac{\log{\left(\left|{\cos{\left(x \right)}}\right| \right)}}{x}\right)}{x} - \frac{\sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} - 1 + \frac{2 \sin{\left(x \right)}}{x \cos{\left(x \right)}} + \frac{\log{\left(\cos^{2}{\left(x \right)} \right)}}{x^{2}}\right) \left|{\cos{\left(x \right)}}\right|^{\frac{2}{x}}}{x}$$
Третья производная
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/ / / 2 \ \ / 2 2 2 \ / 2 / 2 \ \\
| |log\cos (x)/ 2*sin(x)| |2*log(|cos(x)|) cos(x)*sign(cos(x)) sign (cos(x))*sin (x) 2*sin (x)*DiracDelta(cos(x)) 2*sign(cos(x))*sin(x)| 2 / / 2 \ \ /log(|cos(x)|) sign(cos(x))*sin(x)\ | sin (x) log\cos (x)/ 2*sin(x)||
2 | |------------ + --------|*|--------------- - ------------------- - --------------------- + ---------------------------- + ---------------------| /log(|cos(x)|) sign(cos(x))*sin(x)\ |log\cos (x)/ 2*sin(x)| 4*|------------- + -------------------|*|1 + ------- - ------------ - --------||
- | / 2 \ 3 \ x cos(x) / | 2 |cos(x)| 2 |cos(x)| x*|cos(x)| | 2*|------------- + -------------------| *|------------ + --------| 2 \ x |cos(x)| / | 2 2 x*cos(x)||
x |3 3*log\cos (x)/ 2*sin(x) 2*sin (x) \ x cos (x) / 6*sin(x) \ x |cos(x)| / \ x cos(x) / 3*sin (x) \ cos (x) x /|
2*|cos(x)| *|- - -------------- - -------- - --------- - ------------------------------------------------------------------------------------------------------------------------------------------------ - --------- - ------------------------------------------------------------------ + --------- + -------------------------------------------------------------------------------|
|x 3 cos(x) 3 x 2 2 2 x |
\ x cos (x) x *cos(x) x x*cos (x) /
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x
$$\frac{2 \left(- \frac{\left(\frac{2 \sin{\left(x \right)}}{\cos{\left(x \right)}} + \frac{\log{\left(\cos^{2}{\left(x \right)} \right)}}{x}\right) \left(\frac{2 \sin^{2}{\left(x \right)} \delta\left(\cos{\left(x \right)}\right)}{\left|{\cos{\left(x \right)}}\right|} - \frac{\sin^{2}{\left(x \right)} \operatorname{sign}^{2}{\left(\cos{\left(x \right)} \right)}}{\cos^{2}{\left(x \right)}} - \frac{\cos{\left(x \right)} \operatorname{sign}{\left(\cos{\left(x \right)} \right)}}{\left|{\cos{\left(x \right)}}\right|} + \frac{2 \sin{\left(x \right)} \operatorname{sign}{\left(\cos{\left(x \right)} \right)}}{x \left|{\cos{\left(x \right)}}\right|} + \frac{2 \log{\left(\left|{\cos{\left(x \right)}}\right| \right)}}{x^{2}}\right)}{x} + \frac{4 \left(\frac{\sin{\left(x \right)} \operatorname{sign}{\left(\cos{\left(x \right)} \right)}}{\left|{\cos{\left(x \right)}}\right|} + \frac{\log{\left(\left|{\cos{\left(x \right)}}\right| \right)}}{x}\right) \left(\frac{\sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1 - \frac{2 \sin{\left(x \right)}}{x \cos{\left(x \right)}} - \frac{\log{\left(\cos^{2}{\left(x \right)} \right)}}{x^{2}}\right)}{x} - \frac{2 \cdot \left(\frac{2 \sin{\left(x \right)}}{\cos{\left(x \right)}} + \frac{\log{\left(\cos^{2}{\left(x \right)} \right)}}{x}\right) \left(\frac{\sin{\left(x \right)} \operatorname{sign}{\left(\cos{\left(x \right)} \right)}}{\left|{\cos{\left(x \right)}}\right|} + \frac{\log{\left(\left|{\cos{\left(x \right)}}\right| \right)}}{x}\right)^{2}}{x^{2}} - \frac{2 \sin^{3}{\left(x \right)}}{\cos^{3}{\left(x \right)}} - \frac{2 \sin{\left(x \right)}}{\cos{\left(x \right)}} + \frac{3 \sin^{2}{\left(x \right)}}{x \cos^{2}{\left(x \right)}} + \frac{3}{x} - \frac{6 \sin{\left(x \right)}}{x^{2} \cos{\left(x \right)}} - \frac{3 \log{\left(\cos^{2}{\left(x \right)} \right)}}{x^{3}}\right) \left|{\cos{\left(x \right)}}\right|^{\frac{2}{x}}}{x}$$