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Câu hỏi: Tìm đạo hàm của hàm số sau:
\(y = \left( {x\sin \alpha + \cos \alpha } \right)\left({x\cos \alpha - \sin \alpha } \right).\)
\(y = \left( {x\sin \alpha + \cos \alpha } \right)\left({x\cos \alpha - \sin \alpha } \right).\)
Lời giải chi tiết
\(\begin{array}{l}
y' = \left({x\sin \alpha + \cos \alpha } \right)'\left({x\cos \alpha - \sin \alpha } \right)\\
+ \left({x\sin \alpha + \cos \alpha } \right)\left({x\cos \alpha - \sin \alpha } \right)'\\
= \sin \alpha \left({x\cos \alpha - \sin \alpha } \right)\\
+ \left({x\sin \alpha + \cos \alpha } \right)\cos \alpha \\
= x\sin \alpha \cos \alpha - {\sin ^2}\alpha \\
+ x\sin \alpha \cos \alpha + {\cos ^2}\alpha \\
= 2x\sin \alpha \cos \alpha + \left({{{\cos }^2}\alpha - {{\sin }^2}\alpha } \right)\\
= x\sin 2\alpha + \cos 2\alpha
\end{array}\)
\(\begin{array}{l}
y' = \left({x\sin \alpha + \cos \alpha } \right)'\left({x\cos \alpha - \sin \alpha } \right)\\
+ \left({x\sin \alpha + \cos \alpha } \right)\left({x\cos \alpha - \sin \alpha } \right)'\\
= \sin \alpha \left({x\cos \alpha - \sin \alpha } \right)\\
+ \left({x\sin \alpha + \cos \alpha } \right)\cos \alpha \\
= x\sin \alpha \cos \alpha - {\sin ^2}\alpha \\
+ x\sin \alpha \cos \alpha + {\cos ^2}\alpha \\
= 2x\sin \alpha \cos \alpha + \left({{{\cos }^2}\alpha - {{\sin }^2}\alpha } \right)\\
= x\sin 2\alpha + \cos 2\alpha
\end{array}\)