Dùng định nghĩa để tính đạo hàm của các hàm số sau:
a) \(f\left( x \right) = \sqrt {4x + 1} \) tại \(x = 2\);
b) \(f\left( x \right) = {x^4}\) tại \(x = - 1\);
c) \(f\left( x \right) = \frac{1}{{x + 1}}\);
d) \(f\left( x \right) = \sqrt[3]{{{x^2} + 1}}\).
Sử dụng kiến thức về định nghĩa đạo hàm để tính: Cho hàm số \(y = f\left( x \right)\) xác định trên khoảng \(\left( {a;b} \right)\) và \({x_0} \in \left( {a;b} \right)\). Nếu tồn tại giới hạn hữu hạn \(\mathop {\lim }\limits_{x \to {x_0}} \frac{{f\left( x \right) - f\left( {{x_0}} \right)}}{{x - {x_0}}}\) thì giới hạn này được gọi là đạo hàm của hàm số f(x) tại \({x_0}\), kí hiệu là \(f’\left( {{x_0}} \right)\) hoặc \(y’\left( {{x_0}} \right)\). Vậy \(f’\left( {{x_0}} \right) = \mathop {\lim }\limits_{x \to {x_0}} \frac{{f\left( x \right) - f\left( {{x_0}} \right)}}{{x - {x_0}}}\)
a) Với bất kì \({x_0} \ge \frac{{ - 1}}{4}\) ta có: \(f’\left( {{x_0}} \right) = \mathop {\lim }\limits_{x \to {x_0}} \frac{{f\left( x \right) - f\left( {{x_0}} \right)}}{{x - {x_0}}} = \mathop {\lim }\limits_{x \to {x_0}} \frac{{\sqrt {4x + 1} - \sqrt {4{x_0} + 1} }}{{x - {x_0}}}\)
\( = \mathop {\lim }\limits_{x \to {x_0}} \frac{{\left( {\sqrt {4x + 1} - \sqrt {4{x_0} + 1} } \right)\left( {\sqrt {4x + 1} + \sqrt {4{x_0} + 1} } \right)}}{{\left( {x - {x_0}} \right)\left( {\sqrt {4x + 1} + \sqrt {4{x_0} + 1} } \right)}} = \mathop {\lim }\limits_{x \to {x_0}} \frac{{4x + 1 - 4{x_0} - 1}}{{\left( {x - {x_0}} \right)\left( {\sqrt {4x + 1} + \sqrt {4{x_0} + 1} } \right)}}\)
\( = \mathop {\lim }\limits_{x \to {x_0}} \frac{{4\left( {x - {x_0}} \right)}}{{\left( {x - {x_0}} \right)\left( {\sqrt {4x + 1} + \sqrt {4{x_0} + 1} } \right)}} = \mathop {\lim }\limits_{x \to {x_0}} \frac{4}{{\left( {\sqrt {4x + 1} + \sqrt {4{x_0} + 1} } \right)}} = \frac{4}{{2\sqrt {4{x_0} + 1} }} = \frac{2}{{\sqrt {4{x_0} + 1} }}\)
Suy ra: \(f’\left( x \right) = \frac{2}{{\sqrt {4x + 1} }}\). Do đó, \(f’\left( 2 \right) = \frac{2}{{\sqrt {4.2 + 1} }} = \frac{2}{3}\)
b) Với bất kì \({x_0}\) ta có: \(f’\left( {{x_0}} \right) = \mathop {\lim }\limits_{x \to {x_0}} \frac{{f\left( x \right) - f\left( {{x_0}} \right)}}{{x - {x_0}}} = \mathop {\lim }\limits_{x \to {x_0}} \frac{{{x^4} - x_0^4}}{{x - {x_0}}} = \mathop {\lim }\limits_{x \to {x_0}} \frac{{\left( {{x^2} + x_0^2} \right)\left( {x - {x_0}} \right)\left( {x + {x_0}} \right)}}{{x - {x_0}}}\)
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\( = \mathop {\lim }\limits_{x \to {x_0}} \left( {{x^2} + x_0^2} \right)\left( {x + {x_0}} \right) = \left( {x_0^2 + x_0^2} \right)\left( {{x_0} + {x_0}} \right) = 2x_0^2.2{x_0} = 4x_0^3\)
Do đó, \(f’\left( x \right) = 4{x^3}\). Suy ra \(f’\left( { - 1} \right) = 4.{\left( { - 1} \right)^3} = - 4\)
c) Với bất kì \({x_0} \ne - 1\) ta có: \(f’\left( {{x_0}} \right) = \mathop {\lim }\limits_{x \to {x_0}} \frac{{f\left( x \right) - f\left( {{x_0}} \right)}}{{x - {x_0}}} = \mathop {\lim }\limits_{x \to {x_0}} \frac{{\frac{1}{{x + 1}} - \frac{1}{{{x_0} + 1}}}}{{x - {x_0}}}\)
\( = \mathop {\lim }\limits_{x \to {x_0}} \frac{{{x_0} + 1 - x - 1}}{{\left( {x - {x_0}} \right)\left( {x + 1} \right)\left( {{x_0} + 1} \right)}} = \mathop {\lim }\limits_{x \to {x_0}} \frac{{ - \left( {x - {x_0}} \right)}}{{\left( {x - {x_0}} \right)\left( {x + 1} \right)\left( {{x_0} + 1} \right)}} = \mathop {\lim }\limits_{x \to {x_0}} \frac{{ - 1}}{{\left( {x + 1} \right)\left( {{x_0} + 1} \right)}}\)
\( = - \frac{1}{{{{\left( {{x_0} + 1} \right)}^2}}}\)
Vậy \(f’\left( x \right) = \frac{{ - 1}}{{{{\left( {x + 1} \right)}^2}}}\)
d) Với bất kì \({x_0}\) ta có:
\(f’\left( {{x_0}} \right) = \mathop {\lim }\limits_{x \to {x_0}} \frac{{f\left( x \right) - f\left( {{x_0}} \right)}}{{x - {x_0}}} = \mathop {\lim }\limits_{x \to {x_0}} \frac{{\sqrt[3]{{{x^2} + 1}} - \sqrt[3]{{x_0^2 + 1}}}}{{x - {x_0}}}\)
\( = \mathop {\lim }\limits_{x \to {x_0}} \frac{{\left( {\sqrt[3]{{{x^2} + 1}} - \sqrt[3]{{x_0^2 + 1}}} \right)\left( {\sqrt[3]{{{{\left( {{x^2} + 1} \right)}^2}}} + \sqrt[3]{{\left( {{x^2} + 1} \right)\left( {x_0^2 + 1} \right)}} + \sqrt[3]{{{{\left( {x_0^2 + 1} \right)}^2}}}} \right)}}{{\left( {x - {x_0}} \right)\left( {\sqrt[3]{{{{\left( {{x^2} + 1} \right)}^2}}} + \sqrt[3]{{\left( {{x^2} + 1} \right)\left( {x_0^2 + 1} \right)}} + \sqrt[3]{{{{\left( {x_0^2 + 1} \right)}^2}}}} \right)}}\)
\( = \mathop {\lim }\limits_{x \to {x_0}} \frac{{{x^2} + 1 - x_0^2 - 1}}{{\left( {x - {x_0}} \right)\left( {\sqrt[3]{{{{\left( {{x^2} + 1} \right)}^2}}} + \sqrt[3]{{\left( {{x^2} + 1} \right)\left( {x_0^2 + 1} \right)}} + \sqrt[3]{{{{\left( {x_0^2 + 1} \right)}^2}}}} \right)}}\)
\( = \mathop {\lim }\limits_{x \to {x_0}} \frac{{\left( {x - {x_0}} \right)\left( {x + {x_0}} \right)}}{{\left( {x - {x_0}} \right)\left( {\sqrt[3]{{{{\left( {{x^2} + 1} \right)}^2}}} + \sqrt[3]{{\left( {{x^2} + 1} \right)\left( {x_0^2 + 1} \right)}} + \sqrt[3]{{{{\left( {x_0^2 + 1} \right)}^2}}}} \right)}}\)
\( = \mathop {\lim }\limits_{x \to {x_0}} \frac{{x + {x_0}}}{{\sqrt[3]{{{{\left( {{x^2} + 1} \right)}^2}}} + \sqrt[3]{{\left( {{x^2} + 1} \right)\left( {x_0^2 + 1} \right)}} + \sqrt[3]{{{{\left( {x_0^2 + 1} \right)}^2}}}}} = \frac{{2{x_0}}}{{3\sqrt[3]{{{{\left( {x_0^2 + 1} \right)}^2}}}}}\)
Vậy \(f’\left( x \right) = \frac{{2x}}{{3\sqrt[3]{{{{\left( {{x^2} + 1} \right)}^2}}}}}\)