Bài 2/74:
a: \(\lim_{x\to x_0}f\left(x\right)=\lim_{x\to x_0}\frac{\left(2x^2+x+1\right)-\left(2x_0^2+x_0+1\right)}{x-x_0}\)
\(=\lim_{x\to x_0}\frac{2\left(x^2-x_0^2\right)+\left(x-x_0\right)}{x-x_0}=\lim_{x\to x_0}\frac{2\left(x-x_0\right)\left(x+x_0\right)+\left(x-x_0\right)}{x-x_0}\)
\(=\lim_{x\to x_0}2\left(x+x_0\right)+1=2\cdot2\cdot x_0+1=4\cdot x_0+1\)
=>f'(x)=4x+1
Bài 3/74:
a: \(\lim_{x\to x_0}f\left(x\right)=\lim_{x\to x_0}\frac{x^3+1-\left(x_0^3+1\right)}{x-x_0}\)
\(=\lim_{x\to x_0}\frac{x^3-x_0^3}{x-x_0}=\lim_{x\to x_0}x^2+x\cdot x_0+x_0^2=x_0^2+x_0\cdot x_0+x_0^2=3\cdot x_0^2\)
=>f'(x)=\(3x^2\)
b: \(\lim_{x\to x_0}f\left(x\right)=\lim_{x\to x_0}\frac{\sqrt{x^2+1}-\sqrt{x_0^2+1}}{x-x_0}\)
\(=\lim_{x\to x_0}\frac{x^2-x_0^2}{\left(\sqrt{x^2+1}+\sqrt{x_0^2+1}\right)\left(x-x_0\right)}=\lim_{x\to x_0}\frac{x+x_0}{\left(\sqrt{x^2+1}+\sqrt{x_0^2+1}\right)}=\frac{x_0+x_0}{2\cdot\sqrt{x_0^2+1}}=\frac{x_0}{\sqrt{x_0^2+1}}\)
=>\(f^{\prime}\left(x\right)=\frac{\left(x^2+1\right)^{\prime}}{2\sqrt{x^2+1}}=\frac{2x}{2\cdot\sqrt{x^2+1}}=\frac{x}{\sqrt{x^2+1}}\)
