Tangent Lines – The Derivative Function

We’ll be unpacking the formula for tangent lines. So the only example I’m going to show you is the graph of the function f(x) = x^2. We will focus on three notional points on the graph. I will not be naming these points but I will label them later when I label the x and y axes. And here we see the graph of y = x squared. The expression f(a) represents a number, and what is that number? It is the slope of the tangent line to the graph of y = x at the point (x = a). The slope of that little nugget of a line there is equivalent to the acceleration of an object. I only drew part of the line because I don’t want to make the figure too cluttered. If I extended that line in all directions, it would be tangent with the graph We are about to calculate it. But before that, let’s think: should that be positive or negative? That’s positive, since the slope of that little line segment is greater than zero. Therefore, whatever f prime of a is, it better be positive. It is a good idea to perform a reality check before you calculate an answer. Errors can often be caught before they are submitted. Let’s say we started with x = b instead. So we ask, what’s f prime of b? That is, what’s the slope of this tangent line? It should be positive and greater than f prime of a. In other words, whatever we calculate for f prime of b must be bigger than what we calculated for f prime of a–which itself should be greater than 0. On the other hand, if we calculate f prime for c, the slope of the little nugget of a tangent line should be negative because it’s going down. But it must be true that 0 is greater than f prime of c, whatever it is. Okay, so now we’re going to explicitly evaluate this expression. As a warning, much of what is learned in calculus involves lots of techniques for evaluating derivatives. But it is actually quite instructive to work with this definition explicitly. So we will derive the formula for the derivative of f(x) = ax, and let’s compute. So we have f prime of a = lim h goes to 0, and f(a) + h minus f(a) over h. That is always true, but in this case we have a formula for fax). Let’s take the limit as h goes to 0, divided by h. So, what is f(a)? If f(x) = ×2, then f is just a machine that transforms an input into an output of x2. Therefore, f(a) is a2. I’m going to subtract an a2. However, if they input a + h into the machine, what does it do with that input? It squares the entire quantity and outputs a + h squared. Let’s consider the limit as h goes to 0 of (ath) squared – a squared over h. For the moment, don’t be scared of this limit; we will discuss it further later. The limit is equal to a squared + 2ah + h squared when h goes to 0. If we expand this out, we have a squared over h plus 2ah over h plus h squared over h. Good, these a squareds cancel. I can factor out an h to make the limit of h going to 0 of h times 2a + h divided by h. It’s just algebra, nothing too scary if you’re familiar with the basics of algebra. No need to worry. Notice that I have an h on the top and the bottom of the fraction. In algebra class, we know what to do with that. It doesn’t really matter what the h is. We can cancel it out, which leaves us with 2a + h as the limit as h goes to 0. So far, we have avoided discussing what limits are. However, at this point we must think about it. This is equal to, if h is small, 2a plus a small number. As h goes to 0, says if h gets smaller and smaller, what is this approaching? We can essentially set h equal to O here because it’s going away. Since it’s equal to 2a. In other words, my conclusion is that when a is any real number, the derivative of fat the point x = a is 2a. This is an interesting result; let’s see if it makes sense. So, the fact that a was positive tells us that a 2 times a is positive. This is great since a wasn’t really special. In the given case, f prime (b) equals 2b. In other words, “monkey see, monkey do” can be applied to plug in the a for b and so on. As you move from A to B, the derivative of the tangent line at that point gets more and more positive as you go, which makes sense. On the other hand, if c is negative, then f prime of c will also be negative. This makes sense because the slope of a line is always positive for positive values of x. However, if we make c more and more negative, we can see that the slope of f gets steeper and steeper, which also makes sense because slopes rise as you move further to the right along a line. Hopefully, I have convinced you that our approach makes sense. Let’s integrate all of these ideas into our next screen. Now I have created a blue line that represents the graph of f(x) = ×2, rather than telling you an arbitrary reference point. I claim that the equation of this line is y = 2x, but I’ve written this expression f prime (x) = 2x. So what does this mean? Normally, we think of a derivative as a number–the slope of a line at a particular point. But here, we’re talking about a derivative function. The derivative of x squared is the slope of the tangent line to the graph of y=x 2 at the point (x,y)= × . The derivative f prime (x) is an input-output machine that takes in as input x and returns as output the slope of the tangent line to the graph of f(x) = ×2, at the input value x. Let us consider why this is so. If we look at the function f prime of x = 2x, what is true about that line? As x gets larger and larger, 2x also gets larger and larger. And this is right, is it not? Thus, if I take this value here, where that hits the blue line, then that number is supposed to tell me the slope of the tangent line at that point. If I took this value, where it intersects with the blue line, that number is supposed to tell me the slope of the tangent line at that point. That number is higher than that number, and so that slope is more steeply positive than that slope. When x is negative, the blue line gives you a negative value. As you make x more and more negative, it gives you more and more negative values, which is great if x is right here. The y value at which the blue line intersects the curve should be the slope of the tangent line to this curve. It makes sense that it’s negative, because it’s pointing steeply down. Notice that if we look at this equation and take it literally. f prime of 0 = 2 times O, which is O. This means that when x = 0, the slope of the tangent line to this curve is 0. The derivative of f(x) = 2x is 0. This means that the slope of the tangent line to the curve at any point is equal to 2 times the value of x at that point. If we follow along this tangent line, it starts off being really negative, which means the graph is pointing way down. As I move toward the origin, the slope of the curve decreases. It becomes flatter and flatter until it is horizontal. As I move away from the origin to points on the right side of the graph, the slope increases. [Graph]