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Bài 9.26 trang 98 SGK Toán 11 tập 2 - Kết nối tri thức

Xét hàm số luỹ thừa (y = {x^alpha }) với (alpha ) là số thực.

🐷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

Xét hàm số luỹ thừa \(y = {x^\alpha }\) với \(\alpha \) là số thực. a) Tìm tập xác định của hàm số đã cho. b) Bằng cách viết \(y = {x^\alpha } = {e^{\alpha \ln x}}\), tính đạo hàm của hàm số đã cho.

Video hướng dẫn giải

Phương pháp giải - Xem chi tiết

Sử dụng công thức \({\left( {{e^u}} \right)^,} = u'{e^u}\)

Lời giải chi tiết

a) Hàm số luỹ thừa \(y = {x^\alpha }\) với \(\alpha \) là số thực có tập xác định khác nhau, tùy theo \(\alpha \): - Nếu \(\alpha \) nguyên dương thì tập xác định là \(\mathbb{R}\) - Nếu \(\alpha \) nguyên âm hoặc \(\alpha  = 0\) thì tập xác định là \(\mathbb{R}\backslash \left\{ 0 \right\}\) - Nếu \(\alpha \) không nguyên thì tập xác định là \(\left( {0; + \infty } \right)\) b) \(y' = {\left( {{x^\alpha }} \right)^,} = {\left( {{e^{\alpha \ln x}}} \right)^,} = {\left( {\alpha \ln x} \right)^,}{e^{\alpha \ln x}} = \frac{\alpha }{x}{e^{\alpha \ln x}} = \frac{\alpha }{x}.{x^\alpha } = \alpha {x^{\alpha  - 1}}\)

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