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Differentiation from first principles

As h gets small, point B gets closer to point A, and the line joining the two gets closer to the REAL tangent at point A. Calculating the gradient between points A & B is not too hard, and if we let h -> 0 we will be calculating the true gradient.
The point A is at x=3 (originally, but it can be moved!) What is the gradient of the line AB when h = 2? h=1? h=0.1? h=0.01? h=0? (!) The original curve is f(x)=x³. What function would give you the gradient at any point?