In the previous exercise, we discussed the idea of the derivative at a point as the instantaneous rate of change at that point. The idea behind the definition is that by finding how the function changes over increasingly small intervals starting at a point, we can understand how a function changes exactly at that point. The formal expression of this concept is written as:
An equivalent interpretation is that the derivative is the slope of the tangent line at a point. A tangent line at a point is the line that touches the function at that point. Just as the slope of a line describes how a line changes, the slope of the tangent line describes how a “curvier” function changes at a specific point. The visual below shows what a tangent line looks like: In the applet to the right, you can play around with the tangent line and see how it changes as you move to different values of x.
Our discussion of taking increasingly smaller intervals in the last exercise gives us a way to calculate the derivative of a function at a point, but this would be a laborious process if we always had to find the derivative at a point this way.
Fortunately, many common functions have corresponding derivative functions. Derivative functions are often denoted f’(x) (read f prime of x) or df/dx (read derivative of f at x). If f(x) represents the function value at a point x, the corresponding derivative function f’(x) represents how a function is changing at the same point x. Essentially, it saves us from having to draw a tangent line or take a limit to compute the derivative at a point. In later exercises, we will see a number of different rules at our disposal that allow us to calculate the derivatives of common functions.