A derivative measures how fast a function is changing at any given point. Intuitively, it is the slope of the tangent line to the curve at that point. If you zoom in far enough on any smooth curve, it starts to look like a straight line — and the slope of that line is the derivative.
From Secant to Tangent
Start with two points on a curve. The line connecting them is called a secant line, and its slope is the average rate of change. As you move the two points closer together, the secant line approaches the tangent line. The slope of the tangent line at a single point is the derivative.
The Derivative as a Function
The derivative f'(x) is itself a function — it gives the slope of f at every x. For example, if f(x) = x², then f'(x) = 2x. At x = 3, the slope is 6. At x = 0, the slope is 0 (the vertex of the parabola).
What the Sign of f'(x) Tells You
When f'(x) > 0, the function is increasing. When f'(x) < 0, the function is decreasing. When f'(x) = 0, you are at a critical point — possibly a local maximum or minimum.
Explore Derivatives Visually
In VIZMath Calculus, the Derivative Explorer shows a draggable tangent line on any function. As you move the point along the curve, the slope updates in real time and the derivative curve is drawn simultaneously.