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Giải bài 9.38 trang 65 sách bài tập toán 11 - Kết nối tri thức với cuộc sống

Cho hàm số \(y = {e^x}{\rm{cos}}x\). Đẳng thức đúng là

Tổng hợp đề thi học kì 2 lớp 11 tất cả các môn - Kết nố🦄i tri thức

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

Cho hàm số \(y = {e^x}{\rm{cos}}x\). Đẳng thức đúng là

A. \(y'' - 2y' - 2y = 0\).

B. \(y'' - 2y' + 2y = 0\).

C. \(y'' + 2y' - 2y = 0\).

D. \(y'' + 2y' + 2y = 0\).

Phương pháp giải - Xem chi tiết

Áp dụng quy tắc tính đạo hàm \({\left( {uv} \right)^\prime } = u'v + v'u\)

Lời giải chi tiết

\(y = {e^x}{\rm{cos}}x \Rightarrow y' = {e^x}{\rm{cos}}x - {\mathop{\rm s}\nolimits} {\rm{inx}}{e^x} = \left( {{\rm{cos}}x - {\mathop{\rm s}\nolimits} {\rm{inx}}} \right){e^x} = {\rm{cos}}x{e^x} - {\mathop{\rm s}\nolimits} {\rm{inx}}{e^x}\)\(y' = y - {\mathop{\rm s}\nolimits} {\rm{inx}}{e^x}\,\, \Rightarrow {\mathop{\rm s}\nolimits} {\rm{inx}}{e^x} = y - y'(1)\)\(y'' = {\left( {y - {\mathop{\rm s}\nolimits} {\rm{inx}}{e^x}} \right)^\prime } = y' - {\mathop{\rm s}\nolimits} {\rm{inx}}{e^x} - c{\rm{os}}x{e^x}\)\( \Rightarrow y'' = y' - {\mathop{\rm s}\nolimits} {\rm{inx}}{e^x} - y\,\,(2)\)

Từ (1) và (2) suy ra \(y'' = y' - \left( {y - y'} \right) - y\,\, \Leftrightarrow y'' - 2y' + 2y = 0\)

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