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Given: f(x) = 4x2 - 9x - 5 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 4(x+h)2+(x+h)-5 - (4x2-9x-5) h h→0 f ' (x) = lim 4(x2+xh+h2)-9(x+h)-5 - (4x2-9x-5) h h→0 f ' (x) = lim 4x2+xh+4h2-9x-9h-5 + x2+9x+5 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+