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Given: f(x) = -2x2 + 12x + 7 Problem: Determine the derivative of f(x) using the definition of a derivative. f ' (x) = lim f(x+h) - f(x) h h→0 f ' (x) = lim -2(x+h)2+(x+h)+7 - (-2x2+12x+7) h h→0 f ' (x) = lim -2(x2+xh+h2)+12(x+h)+7 - (-2x2+12x+7) h h→0 f ' (x) = lim -2x2+xh-2h2+12x+12h+7 + x2-12x-7 h h→0 f ' (x) = lim xh+h2+h h h→0 f ' (x) = lim h(x+h+) h h→0 f ' (x) = lim x+h+ h→0 f ' (x) =   x+