In my previous post in this series, I introduced the idea of a derivative, and we realized a number of rules regarding them. In this post, we’re going to give some thought as to what derivatives are, look at them from from a rather different perspective, and realize several more rules regarding them.
When we talk about rates, we mean ratios. Think about the units of the examples we’ve used so far. Meters per second. Gallons per second. Movie seconds per real seconds…
If Bob was to tell you that he traveled 100 meters in 2 minutes, you’d be able to tell me that, on average, he was moving at 50 meters a minute or about 0.83 meters a second. So walking speed. But he might have ran for the first 50 seconds at 2m/s and then stood still for the remainder! The derivative is different from the average rate in that it is instantaneous. It’s how fast Bob’s going at a give point in time! To get this we take the value we get as we shrink the range we’re averaging around.
Now we traditionally represent change in a variable as . For example, change in distance is represented , change in time and so on. So average speed is . Since the derivative is the tiny instantaneous changes being divided, we write it as .
We can say that mathematically: the derivative of with respect to , is:
(h always seemed like a silly variable to use here )
The lim part is a limit. They allow us to look at the value as some variable goes to a certain value. In this case, it just means “look at what happens as becomes small.”
A lot can be said about limits. They’re closely tied to the idea of continuity and are very important in analysis, which is the area of higher mathematics that is built around calculus, providing its foundations and extending it in many interesting ways.
But I don’t think it makes sense to talk about limits at an introductory level. (In particular, I don’t think that it makes sense to talk about limits if you aren’t going to use the words `epsilon’ and `delta’.) So I’ll leave the topic here.
Instead, suppose the derivative of y(x) with respect to x is described by a function f(x). So, . But we could rewrite that as:
Meaning that the change in at is the value of times the tiny change in . This is called a differential form. Let’s express the rules we’ve come up with in this manner:
Constant Multiple Rule: =
That may seem a bit cumbersome, but it is actually very nice to work with. As an example, lets determine
By sum rule:
By constant multiple rule:
By chain rule and the trigonometric derivative identity:
And by constant multiple rule again:
Yay! And if we divide by , we get the derivative (instead of the differential), .
Now let’s get back to our “derivative rules” from the differential form perspective.
We can imagine variable as a line and the differential as a dot on the end:
Consider sum rule.We can visualize it as:
In general, you can visualize the differential of a function by imagine that function with the input pushed forward by a dot (representing the differential) and then subtracting the function without the dot.
Let’s consider :
What about division? Let . Then
Traditionally we write this as:
Now, going back to product rule (), if you consider the scenario where both and are , it is clear that , but similar results work in higher dimensions:
Now the final “traditional rule” is the fact that the derivative of is itself. Let’s step back for a second.
Imagine a function that is 1 at 0 and is its own derivative. If you walk backwards, you approach but never touch 0, because the derivative goes to 0 as the function does. On the other hand, the function gets steeper and stepper going forward. If you know what looks like, you know it looks similar to this.
And in fact, lets consider an intuitive example of something that decreases at a rate proportional to itself: the decay of radioactive elements. We all know that they have a half-life, a period of time in which half of the will decay. It is clear the derivative is a negative multiple of the present value, , since the number you loose is proportional to the number you have. And if you think about it in terms of half-life, you realize that half life is an exponential function, something of the form .
But it is very difficult to see that the base of the exponential that is its own derivative is e. I recommend just accepting that as the definition of , or perhaps even the function . Still, if you want to convince yourself of it, consider as ; from here you can apply an argument similar to the one for power rule.
And that’s it! We’re done with derivatives. That’s the content of half a course in Calculus, right there! Let’s do an example of it: calculate . Try and do it by yourself, with out looking at the solution! Feel free to look back at the rules.
First we apply “power rule”, (and also chain rule):
Now by sum rule:
Then by constant rule
Now let’s take a minute to simplify
And since the derivative of the exponential function is itself:
And we’re done!
In the next post in this series, we’ll talk about doing the opposite of differentiation: integration.
I’m really not happy with how I handled exponentials, but I think otherwise this post was pretty good.