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Giải mục 2 trang 83 SGK Toán 11 tập 2 - Kết nối tri thức

Tính đạo hàm của hàm số (y = - {x^2} + 2x + 1) tại điểm ({x_0} = - 1.)

🤪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

Tính đạo hàm của hàm số \(y =  - {x^2} + 2x + 1\) tại điểm \({x_0} =  - 1\).

Video hướng dẫn giải

Phương pháp giải - Xem chi tiết

\(f'\left( {{x_0}} \right) = \mathop {\lim }\limits_{x \to {x_0}} \frac{{f\left( x \right) - f\left( {{x_0}} \right)}}{{x - {x_0}}}\) nếu tồn tại giới hạn hữu hạn \(\mathop {\lim }\limits_{x \to {x_0}} \frac{{f\left( x \right) - f\left( {{x_0}} \right)}}{{x - {x_0}}}\).

Lời giải chi tiết

\(f'\left( { - 1} \right) = \mathop {\lim }\limits_{x \to - 1} \frac{{f\left( x \right) - f\left( { - 1} \right)}}{{x - \left( { - 1} \right)}}\) \( = \mathop {\lim }\limits_{x \to {\rm{\;}} - 1} \frac{{\left( { - {x^2} + 2x + 1} \right) - \left( { - {{( - 1)}^2} + 2.( - 1) + 1} \right)}}{{x + 1}}\) \( = \mathop {\lim }\limits_{x \to {\rm{\;}} - 1} \frac{{\left( { - {x^2} + 2x + 1} \right) + 2}}{{x + 1}} = \mathop {\lim }\limits_{x \to {\rm{\;}} - 1} \frac{{ - {x^2} + 2x + 3}}{{x + 1}}\) \( = \mathop {\lim }\limits_{x \to {\rm{\;}} - 1} \frac{{\left( {x + 1} \right)\left( {3 - x} \right)}}{{x + 1}} = \mathop {\lim }\limits_{x \to {\rm{\;}} - 1} \left( {3 - x} \right) = 3 - \left( { - 1} \right) = 4\). Vậy \(f'\left( { - 1} \right) = 4\).

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