What does the derivative of a function give at a chosen point on its graph?

1. The average gradient between two points on a curve is the gradient of the secant joining them. 2. Slide the second point closer and closer to the first and the secant turns towards the tangent. 3. The limit of that process gives the gradient of the tangent at the single point. 4. That limit is exactly what the derivative calculates. 5. The secant needs two points, so it cannot be what a derivative at one point returns. _Source: Siyavula Mathematics Grade 12 (Everything Maths, CC BY 4.0), Chapter 6: Differential calculus_