How are derivatives related to the slope of a curve at one point?

In mathematics, the derivative of a function at a specific point is geometrically interpreted as the slope of the tangent line to the curve at that point.12

While the slope of a straight line is a constant ratio of "rise over run," curves do not have a constant slope. To determine the slope at a single point on a curve, one cannot simply divide by zero, as this results in a meaningless expression. Instead, calculus employs a limiting process. By choosing a second point on the curve and calculating the average slope over an interval, one can observe the value that the ratio approaches as the interval size ($h$) becomes arbitrarily small, effectively approaching zero without ever reaching it.13

The derivative is defined as the limit of this ratio as $h$ approaches 0. This value represents the instantaneous rate of change of the function at that specific point. Historically, early calculus relied on infinitesimals—quantities smaller than any finite number but not zero—to describe these ratios, though modern mathematics uses the rigorous framework of limits to define them.124