Chapter 14 - A Preview of Calculus: The Limit, Derivative, and Integral of a Function - Chapter Review - Review Exercises - Page 907: 35

$7$

Factor each polynomial: $$f'(1)\\=\lim_{x\to 1}\frac{f(x)-f(1)}{x-1}\\=\lim_{x\to 1}\frac{2x^2+3x+2-(2(1)^2+3(1)+2)}{x-1}\\=\lim_{x\to 1}\frac{2x^2+3x-5}{x-1}\\=\lim_{x\to 1}\frac{(x-1)(2x+5)}{x-1}.$$ Cancel the common factors: $$\lim_{x\to 1}2x+5\\=2(1)+5=7$$