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Given: f(x) = 4x2 + 3x + 6 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)+6 - (4x2+3x+6) h h→0 f ' (x) = lim 4(x2+xh+h2)+3(x+h)+6 - (4x2+3x+6) h h→0 f ' (x) = lim 4x2+xh+4h2+3x+3h+6 + x2-3x-6 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+