Подробное решение
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Не могу найти шаги в поиске этой производной.
Но производная
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Теперь упростим:
Ответ:
1
-------
2 / 2 \
cos (x) | 1 + tan (x) 2*log(tan(x))*sin(x)|
(tan(x)) *|-------------- + --------------------|
| 2 3 |
\cos (x)*tan(x) cos (x) /
$$\left(\frac{2 \log{\left(\tan{\left(x \right)} \right)} \sin{\left(x \right)}}{\cos^{3}{\left(x \right)}} + \frac{\tan^{2}{\left(x \right)} + 1}{\cos^{2}{\left(x \right)} \tan{\left(x \right)}}\right) \tan^{\frac{1}{\cos^{2}{\left(x \right)}}}{\left(x \right)}$$
/ 2 \
1 | / 2 \ |
------- | |1 + tan (x) 2*log(tan(x))*sin(x)| 2 |
2 | |----------- + --------------------| / 2 \ 2 / 2 \ |
cos (x) | 2 \ tan(x) cos(x) / \1 + tan (x)/ 6*sin (x)*log(tan(x)) 4*\1 + tan (x)/*sin(x)|
(tan(x)) *|2 + 2*tan (x) + 2*log(tan(x)) + ------------------------------------- - -------------- + --------------------- + ----------------------|
| 2 2 2 cos(x)*tan(x) |
\ cos (x) tan (x) cos (x) /
---------------------------------------------------------------------------------------------------------------------------------------------------------
2
cos (x)
$$\frac{\left(2 \tan^{2}{\left(x \right)} + \frac{6 \log{\left(\tan{\left(x \right)} \right)} \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 2 \log{\left(\tan{\left(x \right)} \right)} + \frac{\left(\frac{2 \log{\left(\tan{\left(x \right)} \right)} \sin{\left(x \right)}}{\cos{\left(x \right)}} + \frac{\tan^{2}{\left(x \right)} + 1}{\tan{\left(x \right)}}\right)^{2}}{\cos^{2}{\left(x \right)}} - \frac{\left(\tan^{2}{\left(x \right)} + 1\right)^{2}}{\tan^{2}{\left(x \right)}} + \frac{4 \left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)}}{\cos{\left(x \right)} \tan{\left(x \right)}} + 2\right) \tan^{\frac{1}{\cos^{2}{\left(x \right)}}}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$$
/ / 2 \ \
| 3 / 2 \ | / 2 \ 2 / 2 \ | |
1 |/ 2 \ |1 + tan (x) 2*log(tan(x))*sin(x)| | 2 \1 + tan (x)/ 6*sin (x)*log(tan(x)) 4*\1 + tan (x)/*sin(x)| |
------- ||1 + tan (x) 2*log(tan(x))*sin(x)| 2 3 3*|----------- + --------------------|*|2 + 2*tan (x) + 2*log(tan(x)) - -------------- + --------------------- + ----------------------| 2 |
2 ||----------- + --------------------| / 2 \ / 2 \ / 2 \ \ tan(x) cos(x) / | 2 2 cos(x)*tan(x) | / 2 \ 3 / 2 \ 2 / 2 \|
cos (x) |\ tan(x) cos(x) / 4*\1 + tan (x)/ 2*\1 + tan (x)/ / 2 \ 6*\1 + tan (x)/ \ tan (x) cos (x) / 12*\1 + tan (x)/*sin(x) 16*log(tan(x))*sin(x) 24*sin (x)*log(tan(x)) 6*\1 + tan (x)/ *sin(x) 18*sin (x)*\1 + tan (x)/|
(tan(x)) *|------------------------------------- - ---------------- + ---------------- + 4*\1 + tan (x)/*tan(x) + --------------- + ---------------------------------------------------------------------------------------------------------------------------------------- + ----------------------- + --------------------- + ---------------------- - ----------------------- + ------------------------|
| 4 tan(x) 3 tan(x) 2 cos(x) cos(x) 3 2 2 |
\ cos (x) tan (x) cos (x) cos (x) cos(x)*tan (x) cos (x)*tan(x) /
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
2
cos (x)
$$\frac{\left(4 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \frac{4 \left(\tan^{2}{\left(x \right)} + 1\right)^{2}}{\tan{\left(x \right)}} + \frac{12 \left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)}}{\cos{\left(x \right)}} + \frac{24 \log{\left(\tan{\left(x \right)} \right)} \sin^{3}{\left(x \right)}}{\cos^{3}{\left(x \right)}} + \frac{16 \log{\left(\tan{\left(x \right)} \right)} \sin{\left(x \right)}}{\cos{\left(x \right)}} + \frac{3 \left(\frac{2 \log{\left(\tan{\left(x \right)} \right)} \sin{\left(x \right)}}{\cos{\left(x \right)}} + \frac{\tan^{2}{\left(x \right)} + 1}{\tan{\left(x \right)}}\right) \left(2 \tan^{2}{\left(x \right)} + \frac{6 \log{\left(\tan{\left(x \right)} \right)} \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 2 \log{\left(\tan{\left(x \right)} \right)} - \frac{\left(\tan^{2}{\left(x \right)} + 1\right)^{2}}{\tan^{2}{\left(x \right)}} + \frac{4 \left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)}}{\cos{\left(x \right)} \tan{\left(x \right)}} + 2\right)}{\cos^{2}{\left(x \right)}} + \frac{2 \left(\tan^{2}{\left(x \right)} + 1\right)^{3}}{\tan^{3}{\left(x \right)}} - \frac{6 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \sin{\left(x \right)}}{\cos{\left(x \right)} \tan^{2}{\left(x \right)}} + \frac{18 \left(\tan^{2}{\left(x \right)} + 1\right) \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)} \tan{\left(x \right)}} + \frac{6 \left(\tan^{2}{\left(x \right)} + 1\right)}{\tan{\left(x \right)}} + \frac{\left(\frac{2 \log{\left(\tan{\left(x \right)} \right)} \sin{\left(x \right)}}{\cos{\left(x \right)}} + \frac{\tan^{2}{\left(x \right)} + 1}{\tan{\left(x \right)}}\right)^{3}}{\cos^{4}{\left(x \right)}}\right) \tan^{\frac{1}{\cos^{2}{\left(x \right)}}}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$$