The Derivative and Why You Should Care
The first post in this series discussed the idea of the limit. And we tried to present it in a fairly simple, intuitive manner. We also discussed how a lot of the difficulty people ascribe to to calculus is due to the symbology. Above in the header you will see 4 different ways of writing the first derivative. However don’t be dismayed, just stick with me and we’ll see the derivative is not really that bad. Now with reference to the limit you might ask, what does getting really close to some point of anomaly have to do with derivatives? And what is a derivative anyway? And why should I care about all this math-ese? I suppose to motivate you to read on, I will first speak toward the usefulness of this concept.
The derivative allows us to consider the way one quantity or variable responds with regard to another quantity or variable. For instance, consider distance and speed. The more distance I cover in a period of time, the greater my speed. (For more on the relationship between distance, time, speed and acceleration see this post.) Or how about manufacturing: the less material I waste, the less my costs. Or maybe fuel economy: at a low speed I consume a lot of fuel. At a really high speed I consume a lot of fuel. But at some optimum intermediary speed, I maximize my fuel economy. Or maybe designing a structure for maximum strength with minimum weight. Or an electrical system that delivers adequate power with minimum infrastructure. All of these problems and many more utilize the concept of the derivative to arrive at their solutions.
Now with the hope that you are sufficiently convinced of its usefulness, let’s look at what exactly and derivative is. Again, I want to present it in such a way that you can intuitively grasp the idea. Take a look at the following graph.
Now the red line represents the function or the relationship between the input or independent variable, x, and the output or dependent variable, y. There are two of these key points that we want to look at. First at x=a, the function value is y=f(a). We use generic symbols so it will work for any function at any point. We will take a look at a particular function shortly to try and see this less abstractly, but for now let’s continue on. The second point is at x=a+h we get y=f(a+h). Now suppose we are curious to find the slope of the function curve (red line) at the point (a,f(a)) (the line labeled T). A rough approximation could be made by finding the slope of the line between the two points we mentioned. Recall that the slope of a line is equal to the rise divided by the run. The rise then will be equal to the difference in the two y-values, in particular f(a+h) – f(a). The run will be the difference in the x-values, in particular a+h-a which is equal to h. Therefore the approximate slope could be written as:
Now look at the graph and imagine that the point (a+h, f(a+h)) moved closer and closer and closer to the point (a,f(a)). Wouldn’t this give us a better approximation of the slope at (a, f(a))? I bet you have already noticed what concept I’m alluding to. That’s right, the limit. If we use the limit to bring the two points closer and closer together (in fact we will bring them so close together that the distance between them approaches nothing) we will have the exact slope at the first point. This is how we write that:
Now I promised an example so let’s look at a simple one. Let’s use:
Using the definition above we have:
So just like that we have found the slope or derivative of the function x squared which is 2x (“a” was just used above for a particular point, but it could really be any point “x”). In this same fashion we can find the derivative or slope of any function (provided it’s continuous – more on that later.)
But now you might say, “What does finding slope have to do with all the useful stuff you talked about at first?” We will dig into that next time.