slope, Numerical measure of a line’s inclination relative to the horizontal. In analytic geometry, the slope of any line, ray, or line segment is the ratio of the vertical to the horizontal distance between any two points on it (“slope equals rise over run”). In differential calculus, the slope of a line tangent to the graph of a function is given by that function’s derivative and represents the instantaneous rate of change of the function with respect to change in the independent variable. In the graph of a position function (representing the distance traveled by an object plotted against elapsed time), the slope of a tangent line represents the object’s instantaneous velocity.