Gradient (on a graph as I assume you mean), or the differential of the line's equation (dy/dx which means "the difference in y with respect to a difference in x").
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the rate of change is related to the slope; the higher the slope, the higher the rate. If the line is vertical, that is infinite slope or infinite rate of change which is not possible
the slope
Slope is the change in y (vertical dimension) with the change in x (horizontal dimension). On a Cartesian coordinate system, the slope is equal to infinity for change in x = 0, or a vertical line.
Slope.