Understanding Derivatives

What is a Derivative?

In calculus, the derivative of a function measures how the function's output value changes as its input value changes. Think of it as the instantaneous rate of change at a specific point.

Imagine you are driving a car. Your speed at any exact moment is the derivative of your position with respect to time. If your position is described by a function p(t) (position at time t), then your instantaneous speed is the derivative, often written as p'(t) or dp/dt.

Derivative and the Slope of a Tangent Line

Geometrically, the derivative of a function f(x) at a particular point x = a gives the slope of the tangent line to the graph of the function at that point (a, f(a)).

A tangent line is a straight line that "just touches" the curve at that specific point and has the same direction (slope) as the curve at that point.

• If the derivative is positive at a point, the function is increasing there, and the tangent line slopes upwards.
• If the derivative is negative, the function is decreasing, and the tangent line slopes downwards.
• If the derivative is zero, the function has a horizontal tangent line, often indicating a local maximum, minimum, or inflection point.

This simulation will help you visualize the relationship between a function, its derivative, and the tangent line.

Interactive Simulation

Hover over the chart to see the tangent line at that point.