Concept

A function can be thought of as defining a relationship between two sets of variables. For example, the function f(t) = -16t² + v₀t + h₀ tells us what the height of a falling object is at any given time t. But how can we determine the velocity of that object? In other words, how fast is the height changing as time changes? This idea of rate of change is familiar to us as the "slope" of a function, defined as the change in y over the change in x, or

$$slope = \displaystyle\frac{\bigtriangleup y}{\bigtriangleup x}$$

For a simple linear function such as

f(x) = 3x + 1

the slope is constant everywhere and is equal to 3. But what about the slope of our height function above? What is the speed of the object after some time t?

If this function accurately models height, the slope of the function (or the speed of the object) should change over time due to acceleration from gravity. In order to determine the objects speed we must determine the slope of the function. The tool for this is the derivative.

The derivative of a function f(x), written as f'(x), tells us the rate of change of our function. For any function, the slope of the graph at a point is the slope of the tangent line to the graph at that given point. Determine the slope of the tangent line, and we know the rate of change of the function, whether we are talking about velocity of a falling object or rate of population growth for a bacteria.