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