Derivatives from the Definition
Here we are. Derivatives and their definition. After all those years of Math, Arithmetic, Algebra 1, Geometry, Algebra 2, Pre-Calculus and now Calculus. You're ready for the first big thing in Calculus, the derivative. The derivative is very important. The derivative is something that lets you find slopes or rates of change for practically anything. If you're an engineer, you're going to need to use rates. And if you need to use rates, you're probably going to need to use the derivative.
So, at this point you may have already learned how to use derivatives using the shortcuts. That's great. But I want to go back and have you understand why the derivative works. When you first study this, you do go over it before you learn the shortcuts. And then you probably forget it because, it's kind of difficult to do and it's conceptually difficult. Seeing it a second time, I think it's going to make more sense.
Now remember, derivatives are a slope. Slope, rise over run, that old stuff that you've heard since Algebra 1. To produce the formula that will find you a derivative, we have to start with that plain old Algebra 1 rise over run formula. We're going to start off by looking at a secant line. So, you can pick any spot that you want along the curve and the slope is going to be different at different locations.
For example, if you pick this spot here, the slope that goes with that spot looks like it's +3 or something like that. If you picked one over here, and drew in a tangent line at that spot, you'd have a slope of probably -3.
But to find slope, to find rise over run, you need two sets of x, y coordinates. Like, this one and that one. Two coordinates on the line. That's important. Of course if I connect these up, this is going to be just a horrible approximation for the slope at that spot, because this point is so far away. But we'll deal with that a little bit later.
Now, what we have to do is substitute into this, to put it more in the format that you are going to see in the definition. So, if I call this location here x1, then if this formula is f(x), the y coordinate of that is f(x1). If I call this location x2, then its y coordinate is f(x2). Part of the confusion people have with the derivative formula is this function notation. It's so complex, it makes it look worse than it really is. So if you get panicked just remember, slope, rise over run.
We'll be eventually moving up to this tangent line in just a second. Problem of course with the tangent line is that, you don't have two points. Another point on this line right here actually, isn't down the curve. So the rise over run for this, is off the curve. That's where the calculus is going to come in, dealing with that. Here is the slope formula. I've already substituted into it. Looks kind of crazy, doesn't it? I'll show you where all that came from. Knowing that this is the actual spot where we are going to find the slope, I'm going to call that coordinate x.
If you put that x coordinate into the function, you'll have f(x). Now, we need a second point since we are no longer dealing with the secant line. I'm not going to have a point down here where the coordinates would be f(x2). I've got something that's off the shape. So I'm going to have to draw in a right triangle that's actually off of the curve and I'm going to call this distance right here, change of x. So it's how far over that is. That means that this x coordinate is x plus the change of x.
Here is where it really looks scary and crazy. This number is a single number which has to be substituted into the function so it's f(x + change of x). Give me a second to write that. We're set. We have the coordinates for two points along that line, (x, f(x)) and (x+x, f(x + change in x)). Try to say that fast three times. It's already substituted in. This is y2. That's x2 from the slope formula, minus y1 over x1 from the slope formula. This whole quantity is x2, this whole quantity is y2. We are almost done.
Now, the problem with this triangle is it's too big. That's where your calculus is going to come in. You can make change of x smaller and even smaller. Tiny little triangle in there. In fact you keep making it smaller and smaller until change of x is 0. At that point, both the points, point 1 and 2, are going to be both on a straight line and they are going to be on the curve. Of course, if this is zero you can't divide by zero and that's where limits come in. You might want to check out the limit episode to understand a little bit of this. We are just going to go on though, and finish the formula.
So doing a little bit of simplifying, we're almost done. X plus change of x minus the x. The xs go away and all that's left on the bottom is that change of x. Remember change of x gets smaller and smaller as you make these triangles smaller and smaller, until finally change of x is 0 and you can't divide by 0. That's why we are going to need a limit. That's the final formula. Slope is the limit as x approaches 0 of that whole formula.
Now, that we've developed the formula, let's try a little practice problem. We'll do a manual derivative. Remember, this is just the slope formula, that's all it really is. This is the rise and that is the run after being simplified. So we're going to find the derivative of f(x) equals xÂ². That's one of your old friends, a nice simple little parabola. But since it's not a straight line, the slope isn't the same everywhere, that's why you need a formula.
Slope there is different than the slope there, which is different than the slope there and the change. Well, this is the place where you can really get confused easily, because the result of doing this looks so complicated. We urge you to keep calm with this and realize that when you're substituting for this part, what you are doing is taking THE function and actually just putting x plus change of x into the function and replacing this unit. This gets replaced.
When I was learning Calculus, that was the thing that really got me stuck at that. Where does this come from? What goes into where? Such a mess. Just remember, this is going to get replaced with this after you've done the substitution. Let's do that.
So, I'm going to take x + change of x, and use it to replace the x in the formula x squared. You have (x+change of x) quantity squared minus, f(x) is just xÂ² and that's all over changer of x. Now, we need this limit because remember what you're doing. If you're finding the slope at a particular spot, you have to make the slope triangle so small that change of x, is approaching zero. If you put a zero in here right now, you'd be out of luck because you can't divide by zero. That's why we need the limit.
Well, another thing that can daunt you is, the algebraic manipulations for dealing with these are sometimes kind of long and messy. Again, just keep calm, remember what you're doing. If you get to the point where this xÂ² is gone, you've done your job. Let's see what we got.
This is a binomial. Remember when you square a binomial, you do the first thing squared, plus the first times the last term doubled, and then the last term squared. Let's do that. Binomial shortcut. That is xÂ² plus 2 times, the first term times the last. That's 2x change of x. Plus (change of x) quantity squared. Let me put that in a parenthesis so I can be technically correct that a change in x is just one variable. I need to make this line longer for some of these problems. You might be writing across an entire page, if this is like a third power or a fourth power. Minus that xÂ².
In the denominator we still have the change of x and I should still be writing limit every time on this. Limit is change of x approaches 0. Well, now we've got a little bit of simplifying to do. And that's where the magic comes in. And if it doesn't simplify along the lines of what I'm doing, it means that you need to go back and check your work, or find a different way to simplify it. Look what I've got though. I have xÂ² minus xÂ², that adds up to 0.
And now the numerator doesn't have anything that doesn't have change of x in it. That's where the beauty happens. See, unless you can get it to that point, you can't get rid of this change of x in the denominator. And it has to go away because you can't divide by 0. But now I can do it. Both of these terms have a change in x in them. So I can use this change in x to divide that away. That one divides away. This is 2 powers the change of x, so one of them divides away. Look what we have left. Denominator just has a 1, here we have a 2x and just a single power of change of x is left.
Here it's left again. We have just 2x plus a single power of change of x and the denominator was just 1. So I'm not going to write it in there. Now we are free to substitute. See, change of x is approaching 0. I'm not going to be dividing by it, so I can just put it in. Place that change of x with a 0, all that's left of all that, is 2x. That's our derivative.
By this point, I'm certain you've probably learned the slope shortcut, and you wouldn't normally do this. You could just say, slope shortcut. Derivative, you just take the power off the front as a factor, and then make the power 1 less, you get the same thing the fast way.
What you've just learned or more likely relearned, is how derivatives relate to slope. We started off with the basic called the rise over run slope formula, and used that to develop the derivative formula. Derivatives, just are a slope. Now, you probably don't do this much. Again, you've probably learned the shortcut and once you learned the shortcut you never did the manual derivatives again.
To tell the truth on the AP test, it's unlikely they are going to ask you to do the manual derivative like the one we just had in the practice problem. But there is a really good chance that they are going to expect that you will recognize it for what it is. There are some problems that have been seen on the AP test where you're given an expression and they don't tell you what it is. But you have to realize that it really just is the definition of a derivative. And once you recognize that, the problem is very quick and easy.
If you look in the episode about manual limits, you'll find a couple of examples of that. In fact you might want to go and try those right now.