First Derivatives and the Shape of a Graph

The shape of a graph—whether it rises, falls, or levels off—depends on its first derivative. The first derivative gives the slope of the tangent line at any point on a graph.

A positive derivative means the function is increasing and the graph slopes upward. A negative derivative means it is decreasing and the graph slopes downward.

When the derivative is zero or undefined, the point is called a critical point. These points mark locations where the function may change from increasing to decreasing, or vice versa.

They divide the domain into intervals where the first derivative test can be used to find the function’s behavior.

If the derivative is positive before a critical point and negative after, the function reaches a local maximum. If the derivative is negative before the critical point and positive after, the function reaches a local minimum. If the sign does not change, the point is neither a maximum nor a minimum.

First derivatives also help study how diseases spread. A positive derivative shows cases increasing at that moment, while a negative derivative shows cases decreasing and the spread slowing.