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Given: f(x) = -2x2 - 5x + 3 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)+3 - (-2x2-5x+3) h h→0 f ' (x) = lim -2(x2+xh+h2)-5(x+h)+3 - (-2x2-5x+3) h h→0 f ' (x) = lim -2x2+xh-2h2-5x-5h+3 + x2+5x-3 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+