When you are introduced to the study of calculus in your math class, one of the first concepts you will deal with is derivatives. Derivatives do not sound like your typical math functions that you would be familiar with up this point, but they are a core component of calculus and you will work with them in many ways. In this post, I am going to give you a brief introduction to the concept of the derivative, hopefully answering your question “what is a derivative in calculus,” and then I will follow this up in future posts with more detailed examples of using derivatives.
Consider a curve on a graph – any curve will do. Now, pick a point on the graph, and draw a tangent to the graph at that point. Recall that a tangent to a curve is a line which touches the curve at only a single point. This tangent line is a representation of the derivative of that curve at that particular point on the curve. Naturally, every point on your curve will have its own tangent line. Here is a rough example, where I have generated some curve, and the red lines indicate the tangent lines to the curve at the points where the lines touch the curve:
By using the derivative of the equation of your curve, you are able to precisely calculate what the tangent line will look like at any point on your curve. In fact, the derivative of your equation will very likely be another equation, and you can graph this new equation out to see the graph of the derivative, which itself can be interpreted and used in many ways in calculus. You can even find the derivative of the derivative, etc.
Let’s look at a more practical example to hopefully convey the usefulness and significance of calculating derivatives.
If we consider two points A and B, and we know it takes some time to travel from A to B, we can calculate the “average” velocity of this trip by dividing the distance between the two points by the time taken to travel between them. This should be a familiar notion. However, consider this: if we are driving in a car from home to the market, then we do not have the same velocity for the entire trip. We have to start accelerating from being parked, and then decelerate when we come to stop signs, and then accelerate again, and decelerate to a stop when we reach our destination. You can calculate the average velocity of this trip in the way I mentioned above, but the “instantaneous” velocity at any particular point of the trip might be what you are more interested in. How might you go about doing this? Find the derivative! A graph of our trip, where we plot distance travelled against time, might be useful in this case. We can determine the tangents to this curve at any particular points. We know from previous graphing experience that these tangent lines have a slope of rise over run, and distance over time (the y and x axes on this graph) is velocity. So therefore, the tangent line (specifically, its slope) at any particular point on our curve actually represents the instantaneous velocity at that point. So, we can find out exactly how fast we are going at any point of our trip. We just demonstrated this graphically, but by calculating the derivative of our “trip equation” means that we can calculate how fast we were going any any point without actually graphing our equation at all!
This is just an introduction to the concept of derivatives in calculus, but in future posts, I’ll go into more details so that hopefully you can become more familiar with them. The concept might be new and different, but once you begin working with them, you will see that they are not that difficult to work with at all.