interpolation – Cubic Spline Inerpolation – Slope Approximation Using Adjacent Points

I am referencing a paper by CJC Kruger entitled “Constrained Cubic Spline Interpolation for Chemical Engineering Applications.” In the paper he uses a the following formula to calculate slope at a certain point based on the two adjacent points,

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My question is, how is he coming up with this equation exactly? It looks kind of like an average of two slopes but I know that isn’t right.

Thank you!