[tex]a)f'(x)= \frac{( x^{2} +x+1)'}{2 \sqrt{ x^{2} +x+1} } = \frac{2x+1}{2 \sqrt{x^2+x+1} } \\ b) [tex]y=mx+n \\ m= \lim_{x \to \infty} \frac{ \sqrt{x^2+x+1} }{x}= \lim_{x \to \infty} \frac{x \sqrt{1+ \frac{1}{x}+ \frac{1}{ x^{2} } } }{x} =1 \\ n= \lim_{x \to \infty} ( \sqrt{x^2+x+1}-x)= \lim_{x \to \infty} \frac{ x^{2} +x+1-x}{ \sqrt{x^2+x+1}+x } = \\ = \lim_{x \to \infty} \frac{x+1}{ \sqrt{x^2+x+1}+x } = \lim_{x \to \infty} \frac{x(1+ \frac{1}{x}) }{x( \sqrt{1+ \frac{1}{x}+ \frac{1}{ x^{2} } }+1) } = \frac{1}{2} \\ y=x+ \frac{1}{2} -asimptota-oblica[/tex]
