Concept of Derivative Function

The Idea Behind Derivatives

Imagine you're riding a bicycle on a hilly road. Sometimes the road is steep, and other times it's flat. The slope of the road changes at every point you pass. In mathematics, the graph of a function can be thought of as this hilly road.

For a straight line, the slope is always the same at every point. However, for a curved line, the slope is constantly changing. A derivative lets us find the precise slope or rate of change at one specific point on a curve.

Gradient of a Secant Line

To understand the concept of a derivative, let's start with something simpler: a secant line (or a cutting line). A secant line is a straight line that intersects a curve at two different points.

Suppose we have a curve from the function $y = f(x)$. We pick two points on that curve, let's call them point and point . Here, (read "delta x") represents a small change in the value of .