Derivative Intuition
See the instantaneous slope of a curve at every point
Summary
The derivative is a function's instantaneous rate of change at a chosen point: the slope of the tangent line drawn there. The function f(x) with its tangent and the derivative f'(x) are shown in separate panels. As you drag the point along the curve (or move the x₀ slider), watch how the slope of the tangent matches, exactly, the height of the point on the derivative graph.
In depth
The derivative, the cornerstone of differential calculus, was developed independently by Leibniz and Newton in the late 17th century. The derivative f'(x₀) of a function f at a point x₀ equals the slope of the tangent line to the curve at that point. Geometrically: for an infinitely small change around x₀, how much does f change? This interactive visualisation shows a set of basic functions (such as x², x³ − 3x, sin x and eˣ) and their derivatives in two separate panels, one above the other: the function f(x) with its tangent line at the chosen point, and the derivative function f'(x). A dashed vertical line at your chosen x₀ links the two graphs: the slope of the tangent equals the y value of the point on the derivative graph exactly. This match is no coincidence: the derivative function is, by definition, "the slope at every x".
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