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Given: f(x) = -3x2 + 8x + 8 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 -3(x+h)2+(x+h)+8 - (-3x2+8x+8) h h→0 f ' (x) = lim -3(x2+xh+h2)+8(x+h)+8 - (-3x2+8x+8) h h→0 f ' (x) = lim -3x2+xh-3h2+8x+8h+8 + x2-8x-8 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+