Bài tập 9 trang 192 SGK Toán 11 NC

Bài tập 9 trang 192 SGK Toán 11 NC

a) Đặt \(f\left( x \right) = y = \frac{1}{{2x - 1}}\)

Với \({x_0} \ne \frac{1}{2}\) ta có:

\(\begin{array}{*{20}{l}}
{f\prime ({x_0}) = \mathop {\lim }\limits_{\Delta x \to 0}  = \frac{{f({x_0} + \Delta x) - f({x_0})}}{{\Delta x}}}\\
{ = \mathop {\lim }\limits_{\Delta x \to 0} \frac{{\frac{1}{{2{x_0} + 2\Delta x - 1}} - \frac{1}{{2{x_0} - 1}}}}{{\Delta x}}}\\
{ = \mathop {\lim }\limits_{\Delta x \to 0} \frac{{ - 2\Delta x}}{{\Delta x(2{x_0} + 2\Delta x - 1)(2{x_0} - 1)}}}\\
{ = \mathop {\lim }\limits_{\Delta x \to 0}  - 2(2{x_0} + 2\Delta x - 1)(2{x_0} - 1)}\\
{ = \frac{{ - 2}}{{{{(2{x_0} - 1)}^2}}}}
\end{array}\)

b) Đặt f(x) = \(y = \sqrt {3 - x} \) 

Với x₀ < 3, ta có:

\(\begin{array}{*{20}{l}}
{f\prime ({x_0}) = \mathop {\lim }\limits_{\Delta x \to 0}  = \frac{{f({x_0} + \Delta x) - f({x_0})}}{{\Delta x}}}\\
{ = \mathop {\lim }\limits_{\Delta x \to 0} \frac{{\sqrt {3 - {x_0} - \Delta x}  - \sqrt {3 - {x_0}} }}{{\Delta x}}}\\
{ = \mathop {\lim }\limits_{\Delta x \to 0} \frac{{ - 1}}{{\sqrt {3 - {x_0} - \Delta x}  - \sqrt {3 - {x_0}} }} = \frac{{ - 1}}{{2\sqrt {3 - {x_0}} }}}
\end{array}\)

-- Mod Toán 11