Understanding this concept is half of Calculus, and it has vast implications
What if I told you that you have had the fundamentals on how to take a derivative since Algebra? You would probably laugh at me and call me crazy, and you would be right. Really though, the formula for a derivative is based on the formula for the slope of a line, m = (y2-y1)/(x2-x1). For the graph given, the slope of the line going through the two points of interest can be found using that slope formula. This is known as a secant line and represents the average rate of change through two points, or simply the average.
Then how does the derivative tie in? The derivative represents a tangent line or instantaneous rate of change. This is what you see on the speedometer of a car.
The speedometer does not tell you the average speed that you have been going over the past 30 minutes or hour. It tells you what speed you are going at right now. How do we find this analytically? We do this by using the familiar concept of limits. We find the limit of the slope equation as x2 approaches x1 in our slope formula. The value of this limit is the slope of the tangent line at that point. This is the same as the derivative.
This is great and all, but what is a derivative actually used for? We already mentioned the speedometer example. Computers use a derivative to find solutions to equations by a trial and error method. One example is the Newton-Raphson Method. Derivatives can also be used to minimize or maximize certain variables in a process known as optimization. Finally, since derivatives show instantaneous rates of change, they can also be used for comparing how two related variables change, such as the volume of a sphere versus its diameter. Derivatives have brought advancements in statistics, physics, economics, and many other fields.