Welcome to Calculus. I'm Professor Ghrist. We're about to begin Lecture 10 on Derivatives. We're about to begin Chapter 2 on Differentiation. Now, you may know how to compute some derivatives. But do you know what they really mean? In this lesson, we'll distinguish between the interpretation and the definition of a derivative. And by the end of the lesson, you'll be seeing derivatives everywhere. In your introduction to calculus, you probably learned derivatives as slopes. Well, you may be surprised that we have not talked very much about slopes in this course thus far. And, that's because slope is an interpretation of the derivative and not what it really means. It is not the definition. In fact, it's a pretty poor interpretation. For example, when you go to multi variable calculus, you're going to look at functions with multiple inputs and multiple outputs. What does a derivative mean for that? Let's say you have a vector field or you're looking at a digital image that consists of many, many variables. Or worse still, you're looking at functions that aren't smooth enough to have a well-defined slope. In all of these contexts derivatives make sense even when slopes do not. We'll consider three different definitions of the derivative. The derivative of f at x equals a is first, the limit as x goes to a of f(x)- f(a) over x- a. This might not be the definition that you recall seeing. But it's a great definition because you can interpret it as the limit of the change in the output over the change in the input as the input tends to a. This definition, among all of the definitions is to me, conceptually the clearest. You have an input and an output, and as you change that input at a certain rate, the derivative tells you at what rate the output is changing. Our second definition should look familiar. The derivative of f at a is the limit as h goes to 0 of f(a + h)- f(a) over h. This too has the interpretation as the limit of the change in output over change in input as the change in input, h, goes to 0. Of course, this is really the same definition. We've simply performed a change of variables, setting h to be equal to x- a, the change in the input. If you substitute x- a in for h above, writing x as a + h, then you'll see that this is really the same definition. Our third definition is a bit different, and it's in terms of the first order variation of the output. And what do I mean by that? Well consider your function evaluated not at a but at a + h. What is that equal to? Well, it's f(a) + some perturbation term, some variation term that depends on h. We say that the derivative of f at a is the constant C, satisfying f(a + h) = f(a) plus C times h + other stuff that is of higher order in h. That is, big O of h squared. This constant in front of the first order term is what we can define the derivative at a to b. Now this is not the definition that you are used to seeing. In fact, there are some problems with this definition. Some people call this a strong derivative, because sometimes it does not exist, even when the true derivative in terms of a limit does. For purposes of this course, we're not gonna worry about that distinction so much. Thinking in terms of a Taylor expansion and using the language of big O to control higher order terms is going to wind up being very illuminative. Let's compare these different definitions in the context of a simple example. The simplest one I can think of is f(x) = x to the n for some positive integer n. In this case, the first two definitions of the derivative of f at a are of the form limit as h goes to 0, f(a +h)- f(a) all over h. Knowing that f is really raising to the nth power allows us to simplify a bit. What happens when we take a + h and raise it to some positive integer? Well the first term is going to be a to the n. The next term is going to be n times a to the n-1 times h. And this comes from the binomial theorem or multiplying out as a long polynomial. What about all of the other terms? I may not have room on the page to write them all. But I know that all of them have powers of h that are at least quadratic. I'm going to compress all of that together and call it big O of h squared, which is exactly what that is. And now we see a, perhaps, familiar computation coming about. The a to the n terms cancel, and everything that is left has some power of h. We can factor that, cancel with the denominator. And we obtain the limit as h goes to 0 of n times a to the n-1 + some terms that are in big O of h. So that as h goes to 0, they vanish, and we are left with the familiar computation of the derivative of x to the n at a. Now, it's worth observing that really that definition of the derivative is the same as the third definition in terms of the variation of the output. If we take our function f and evaluate it at a+h, what do we get? a+h to the n, this has a constant term, a to the n. A first order term, n times a to the n-1 times h + higher order terms in h. That coefficient, the first order term, is the derivative at a. A few other examples will help illustrate this. Let's look at some familiar functions, e to the x, cosine of x, and square root of x. First of all, for e to the x, let's compute the first order variation by evaluating it at x + h. Now, we know that e to the x + h is e to the x times e to the h. And although this is a little bit of circular reasoning, we know what e to the h means. And that's really 1+ h + terms of higher order in h. So, if we expand this out, we see that the constant term is e to the x. The first order term in h is e to the x times h. All other terms are higher order in h. And from this simple computation, we can see that the derivative of e to the x is e to the x. It's the coefficient in front of the first order term in h. Likewise for cosine, what happens when we look at cosine of x + h? Well, we can use the summation formula for cosines and expand this as cosine(x) times cosine (h)- sine(x) times sine(h). And again, using what we know about cosine and sine, we can expand the terms in h and say that cosine of h is 1 + big O of h squared. And sine of h is h + big O of h cubed. Now, if we take these terms and rearrange them a little bit, we get a zeroth order term of cosine of x as we must. The first order term has as its coefficient in front of h, -sine(x), the derivative of cosine of x. For the square root, we have to simplify the square root of x + h. How do we do that? Well, if we factor out a square root of x, what is left over is 1 + h over x, all to the power of one-half, or the square root of it, if you like. I write it in this form so that we can see the binomial series come into play. And using that binomial series gives us an expansion of 1 + one-half times h over x + something in big O of h over x, quantity squared. Now if we expand that out, we see that the zeroth ordered term in h is of course, square root of x. The first order term simplifies to one-half x to the -one-half times h. And from that we see the derivative. Now notice that all of the higher order terms involve a square root of x and something in big O of h squared over x squared. If we want to ignore that and call that big O of h squared, we'd better make sure that x is positive. When x is zero, we have a problem. The derivative does not exist. But as long as x is bounded away from zero, we have a well-defined expansion in terms of h. And we can read off the derivative as 1 over 2 square root of x. There's a lot of notation associated with derivatives. The derivative of the function y = f(x) can be denoted in the following ways. Some of the best notation is df dx or dy dx. But there's some other notation that you might see or use from time to time, including f prime, or y prime, or y dot. Now, why the difference between these? Well, the best notations are those that tell you exactly which variables are changing. dy dx means the rate of change of y with respect to change in x. On the other hand, f prime or y dot can be a bit ambiguous. The dot, in particular, connotes change with respect to time. Sometimes you'll see a differential notation of the form df. This can be helpful. But again, remember which variables are changing. The one thing that you must never do is write something foolish like cancelling the ds to get df, dx = f over x. Whew, no way. Don't get creative with your handwriting on these. Don't use a scripty cursive d, don't use a Greek delta. Don't do anything like that because those symbols have meanings of which you have not yet learned. Don't ever do any of this. Stick to the standard notation, please. Now the most common examples of derivatives wind up involving rates of change with respect to time. For example, in your introductory physics, you've certainly seen velocity or acceleration as a derivative. These are derivatives of position with respect to time. But there are other examples as well. If you study current, current is related to a rate of change of charge with respect to time, flowing through a wire. In chemistry, if you look at reaction rates in a chemical reaction, these are defined as the derivative of say, a concentration of a product or a reactant with respect to time. There are many other examples of derivatives that are not necessarily rates of change with respect to time. If you look at a spring, one can define the spring constant as the rate of change of the force applied with respect to the deflection. The elastic modulus of a stretchy material is a rate of change stress with respect to strain. If you want to know what viscosity is or how slippery a fluid is, well this is defined in terms of quantities like shear stress and the rate of change of the velocity of a sliding fluid with respect to height. And finally, economics consists of all manner of interesting derivatives. For example, if you look at marginal tax rates, these are defined as the rate of change of the amount of tax collected with respect to change in income. All of these are wonderful examples of derivatives. There's so much more to derivatives than slope. Look around you. Do you see something that changes? That is a derivative. Derivatives are ubiquitous, and understanding their proper definition helps us to interpret, find, and then use derivatives. In our next lesson, we'll consider how to quickly compute derivatives.
