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Giải bài 9.26 trang 63 sách bài tập toán 11 - Kết nối tri thức với cuộc sống

Đạo hàm của hàm số \(y = \sqrt {1 + 2{{\sin }^2}x} \) là

🐷Tổng hợp đề thi học kì 2 lớp 11 tất cả các môn - Kết nối tri thức

Toán - Văn - Anh - Lí - Hóa - Sinh
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Đề bài

Đạo hàm của hàm số \(y = \sqrt {1 + 2{{\sin }^2}x} \) là

A. 🐈\(y' = \frac{{\sin 2x}}{{\sqrt {1 + 2{{\sin }^2}x} }}\).

B. ♋\(y' = \frac{{\sin 2x}}{{2\sqrt {1 + 2{{\sin }^2}x} }}\).

C. 🐷\(y' = \frac{{\sin 2x}}{{\sqrt {1 + 2{{\sin }^2}x} }}\).

D. 🅠\(y' = \frac{{\sin x\cos x}}{{2\sqrt {1 + 2{{\sin }^2}x} }}\).

Phương pháp giải - Xem chi tiết

Áp dụng công thức đạo hàm của hàm số lượng giác

\({\left( {{{\sin }^n}u} \right)^\prime } = u'.n.\cos u.{\sin ^{n - 1}}u\)

\({\left( {\sqrt u } \right)^\prime } = \frac{{u'}}{{2\sqrt u }}\)

Lời giải chi tiết

\({\left( {\sqrt {1 + 2{{\sin }^2}x} } \right)^\prime } = \frac{{{{\left( {1 + 2{{\sin }^2}x} \right)}^\prime }}}{{2\sqrt {1 + 2{{\sin }^2}x} }} = \frac{{4\sin x.\cos x}}{{2\sqrt {1 + 2{{\sin }^2}x} }} = \frac{{2\sin x.\cos x}}{{\sqrt {1 + 2{{\sin }^2}x} }} = \frac{{\sin 2x}}{{\sqrt {1 + 2{{\sin }^2}x} }}\)

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