Giải Bài tập 1 trang 41 Toán 11 tập 2 Chân trời

a) $f'(x_{0}) =\lim_{x \to x_{0}}\frac{f(x)-f(x_{{0}})}{x-x_{{0}}}=\lim_{x \to x_{0}}\frac{-x^{2}-(-x_{0}^{2})}{x-x_{0}}$

$=\lim_{x \to x_{0}}\frac{-(x^{2}-x_{0}^{2})}{x-x_{0}}= \lim_{x \to x_{0}}\frac{-(x-x_{0})(x+x_{0})}{x-x_{0}}$

$= \lim_{x \to x_{0}}[-(x+x_{0})] = -(x_{0}+x_{0})=-2x_{0}$

b) $f'(x_{0}) = \lim_{x \to x_{0}}\frac{f(x)-f(x_{{0}})}{x-x_{{0}}}=\lim_{x \to x_{0}}\frac{x^{3}-2x - x_{0}^{3} +2x_{0}}{x-x_{0}}$

$= \lim_{x \to x_{0}}\frac{(x^{3}-x_{0}^{3})-(2x-2x_{0})}{x-x_{0}}$

$= \lim_{x \to x_{0}}\frac{(x-x_{0})(x^{2}+x.x_{0}+x_{0}^{2})-2(x-x_{0})}{x-x_{0}}$

$=\lim_{x \to x_{0}}[(x^{2}+x.x_{0}+x_{0}^{2})-2]$

$=(x_{0}^{2}+x_{0}.x_{0}+x_{0}^{2})-2= 3x_{0}^{2}-2$

c) $f'(x_{0}) = \lim_{x \to x_{0}}\frac{f(x)-f(x_{{0}})}{x-x_{{0}}}=\lim_{x \to x_{0}}\frac{\frac{4}{x}-\frac{4}{x_{0}}}{x-x_{0}}$

$= \lim_{x \to x_{0}}\frac{\frac{4x_{0}-4x}{x.x_{0}}}{x-x_{0}}= \lim_{x \to x_{0}}\frac{-4}{x.x_{0}}$

$=\frac{-4}{x_{0}.x_{0}}=\frac{-4}{x_{0}^{2}}$