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Giải bài 9.4 trang 57 sách bài tập toán 11 - Kết nối tri thức với cuộc sống

Tính đạo hàm của hàm số

Tổng hợp đ🏅ề thi học kì 2 lớp 11 tất c꧃ả các môn - Kết nối tri thức

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

Tính đạo hàm của hàm số a) \(y = a{x^2}\) (\(a\) là hằng số) tại điểm \({x_0}\) bất kì. b) \(y = \frac{1}{{x - 1}}\) tại điểm \({x_0}\) bất kì, \({x_0} \ne 1\).

Phương pháp giải - Xem chi tiết

Để tính đạo hàm của hàm số \(y = f(x)\) tại điểm \({x_0} \in (a;b)\), ta thực hiện theo các bước sau: 1. Tính \(f(x) - f\left( {{x_0}} \right)\). 2. Lập và rút gọn tỉ số \(\frac{{f(x) - f\left( {{x_0}} \right)}}{{x - {x_0}}}\) với \(x \in (a;b),x \ne {x_0}\). 3. Tìm giới hạn \(\mathop {\lim }\limits_{x \to {x_0}} \frac{{f(x) - f\left( {{x_0}} \right)}}{{x - {x_0}}}\).

Lời giải chi tiết

a) \(y'\left( {{x_0}} \right) = \mathop {\lim }\limits_{x \to {x_0}} \frac{{ax_{}^2 - ax_0^2}}{{x - {x_0}}} = \mathop {\lim }\limits_{x \to {x_0}} a\left( {x + {x_0}} \right) = 2a{x_0}\)
b) \(y'\left( {{x_0}} \right) = \mathop {\lim }🔴\limits_{x \to {x_0}} \frac{{\frac{1}{{x - 1}} - \frac{1}{{{x_0} - 1}}}}{{x - {x_0}}} = \mathop {\lim }\limits_{x \to {x_0}} \left[ { - \frac{1}{{\left( {x - 1} \right)\left( {{x_0} - 1} \right)}}} \right] =  - \frac{1}{{{{\left( {{x_0} - 1} \righ♔t)}^2}}},{x_0} \ne 1\)

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