All right, hi, everyone, and welcome back. Today we're going to start our new section called the derivative as a function, but before we do that, let us start with a little bit, perhaps, of a warm-up. The last section, things get a little messy, so we can't do enough problems, so let's just do a little warm-up. Pause the video after I write it. See if you can do it, and then we'll go over the solution together. So here's what I want you to do. Find f of a, so f is some function, a is some number, given that the limit as h goes to 0 of 1 + 8 to the 10th- 1 over h is a derivative. So this is equal to f prime of a for some f and for some a, for some f and some a. Okay, pause the video, this is your warm-up, and see if you can figure this out. Okay, I'll wait. No, I won't, I'm kidding, hopefully you paused the video. Ready, we're going to do the answer quick, quick, quick. All right, here we go. This one is a little backwards, right? So of course, whenever they give you a question, they start to give it to you in reverse so you are fluent in mathematics, going forwards and backwards. In this example, they give you the limit and they ask for the function and the point, and the ones you've seen prior, they gave you the function and the point and ask you to find the derivative, the slope of the tangent line at that value. So we have to remember what the formula is for the slope of the tangent line, what the derivative is. So this is the limit as h goes to 0 of f of a + h minus f of a, but normally, since we're plugging in a point f of a all over h. So that formula, again, hopefully that is written somewhere clear for you to come back to, we're going to use it all the time. Here is the limit definition, so I have to find an f and a value a that matches this expression, the one that's given. So hopefully it's clear that a is 1, right? So it's a + h, 1 + H. The number I'm adding is a, so you can look at this and just tell me that a has to be 1. It's a little trickier, although not too much, to try to find the function that works. So this is a little bit of guess and check, but I think you can see it quickly. So what function, if I plug in 1 + h, raises the whole expression to the 10th, what expression? So my guess would be x to the 10th, that would be my guess. Now, you should check this. This is your check using this part over here, the right part of f of 1. Is f of 1 equal, remember, it has to equal 1? This is a question, if that's my choice, is f of 1 equal to 1? Does it all work? So you pick the function, you guess the function, you check if I actually use this function and plug in my a, do I get back 1? And yes, 1 of the 10th is equal to 1, so we're good to go and I have found the function and the number that is represented by this derivative. So here, it's like Jeopardy, they give you the answer and you have to find the question. All right, so just a little way to see it in reverse. We should be comfortable doing both. Let's do another one, so same thing, same kind of expression. So find a function, so find f, and let's say find f of x, and we'll say we'll find a, so some real number a, given the following. So from f prime of a is equal to the limit as, now be careful here, x goes to 5 of 2x- 32 over x- 5, over x- 5. Now, this is weird because we're kind of pulling it all together, and we're seeing, we're getting a hint of what's to come. If I had given you this crazy limit, you may not have wanted to do the algebra to work this out, 10 times 1 plus h to the 10th. But if you recognize it as a derivative of a function, then I'll be able to use derivatives to find limits. That's kind of weird because derivatives are defined by limits, so it seems circular, and there's a relationship between the two, and that's how we use one to find the other. So this expression is very similar. However, the trick, the catch to this one is it's using the other definition of the limit of derivatives. So remember, we can also define, we call this version number two, you can define the derivative by making a substitution in the original definition as the limit as x approaches some number a of f of x- f of a over x- a. This is a perfectly equivalent way to think of the slope of the tangent line at the point. 0.8 transform just making a substitution of variables and that's the form they're handing it to you in this case. So that's the one. The other one that you should know and recognize it when you see it. When it's in this form, it's pretty easy to just pull off a again. Be careful, the minus signs here, so it's -a -5. So put these two things together and you get the number in question is 5. And then some most just as easy. This is almost easier even though it's using the second version of the form. The function and question you can read that off of the expression as well. So f(x)=2x. As a quick check just to make sure things are going well, let's use the f(a) term here, so check is f(5) =32? That's the question because here's f(a) and the definition, and it has to equal to 32 in the given expression. And you can check 2 to the 5th is in fact equal to 32. So our final answer here going backwards given the limit and then describing the function that it describes, a is equals 5 and f(x) =2 to the x. Okay, so those are just two little warmups that I want you to see and be comfortable with. And again, remind yourself, write it down if you don't have it already what the definition of the derivative is in both its form. Okay, now let's get onto some new content. The derivative of a function. So right now we have write it again, we're going to write this thing a million times. The slope of the tangent, the the derivative of a function is f prime at a point is the limit as h goes to 0 of f(a+h)- f(a), all over h. So a is some number inside of the domain of the function. Maybe this limit exists we've seen examples where it does not. Maybe it does, when it does, it describes the instantaneous rate of change at the point a it describes the slope of the tangent line at point a. And so the idea now is instead of picking it for one particular a so now, replace a specific a with a variable x and it becomes a function. So it looks all the same, so it becomes a limit as h goes to 0 f(x+h) -f(x) all over h. And it becomes a function. So now and this is going to feel that weird, because I'm going to ask you for the derivative of a function. You're going back a function. Is not a satisfying initially to be the answer is 7 that feels good, right? But if I ask you what is the derivative function, you hit me back another function, it's better. Because instead of telling me what the derivative is at a single point with the slope of the tangent line at a single point is you're telling me what this derivative is at every single point. So let's jump right into this with an example. And you'll see what I mean. So let f(x) be 5x cubed. 5x cubed- 8x. What's the derivative as a function? And we can compare this with the graph. So this will be the limit as h goes to 0 of (f(x+h) -f(x)) all over h. Hopefully we're getting good at writing that definition. Here we go. Let's work it out as becomes the limit as h goes to 0, you have to write the limit every time. Do I have to write the limit every time? Yes, you have to write the limit every time. Whenever I see an x in the original function, I replace it with x+h. We're going to use parentheses to be careful,- 8(x+h). So there's the expression with x+ h plugged into it minus the original function. Be careful, use parentheses, and now we have a big fun, no fun, fun algebra expression to solve. Let's come over here and do that so it becomes the limit is h goes to 0 of 5. Now I'm going to do this in my head because I've done this a million times, but if you need to pause the video and work this out, I encourage you to do so. It becomes x cubed, 3x squared h +3xh squared + h cubed. I can do that in my head, not because I'm super smart, but only because I've basically memorize this thing from having done it for so long. But pause the video here. This is usually where students feel, how did you do that? Go off and foil this three times. I don't have the room on the slides to do it, nor do I want to bug you down with algebra. So go check me if you want. The h Can be distributed in -8x-8h. And then this minus sign is going to get brought in as well. So -5x cubed + 8h, and this whole big thing is over h. And this is kind of gross. But calculus is kind of cool, because it always looks worse before it gets better. It's like a bad bruise, just as painful, so stuff cancels, right? Hopefully you can see what's happening. So if there's a 5X cubed here, and that will cancel with the 5X cubed over here, what's left here? There's a 15, where we gotta bring the 5 in, so that's x squared h. That's the only one of its kind. So there's 15h squared h, so I have this term. There's also going to be a positive 15xh squared, and then there's a 5h cubed. So those are fine. What's next? -8X, that will cancel with the positive 8x on the other side, and I'm left with a -8h. And all of that is over h. So stuff is cancelling, always a good sign, and then there's even more to cancel. There's an h on the bottom, and every single term, that will come as squared, has an h in it as well. And so the h in the denominator, the reason why I couldn't evaluate the limit in step one, has cancelled. And hopefully, you're starting to see the pattern for these things, that that's what we're hoping will happen. The last step to do, so what are we left with here? So we have the limit as h goes to 0, there's a 15x squared + 15xh+ 5h squared- 8. That's a beautiful polynomial. I don't like that color, do you like that color? It's a beautiful polynomial. How do we evaluate limits of polynomials? You just plug in h equals 0, so this term goes to 0. This term goes to 0, the 15xh, the 15h squared goes to 0, and my final answer that I'm going to squeeze in here, good thing I got to the end when I did, is 15x squared- 8. 15x squared- 8, that is our answer. You say, what's the answer? Where's the number? No, no no, I gave you a function and I'm giving you back a function. This is me handing you infinitely many answers. So now if you want the derivative at x = 1, instead of going through the whole process for x = 1, I just plug in 1 into my derivative. So the derivative of this function, f(x) is 15x squared- 8, 15x squared- 8. So I can say, I'll plug in at 1, it's 15- 8, so it's 7. We'll say, okay, well what is the derivative? What is the slope of the tangent line at 2? Instead of me doing a whole page of work just for 2, because I had the function, now I have this machine that just tells me the answer to the derivative. So you can plug in 15 * 4- 8 and work that out or 3 or 4. And you can have this whole new function that gives you any value you want. It's such a more powerful tool than just working this out for one single number, one single number, and that's really nice. And with any function, now that I have a new function, I can ask things that I would do of any function. What's the graph? What's the domain? What's the range? And you can graph these things and overlay them. And you can start to see that there's a relationship between the function and the derivative. And we're going to explore this relationship further, but each piece, the function has information about the derivative and the derivative has information about the function. And this has deep consequences, because remember what the derivative is, it's measuring change. So if you can measure the change of something, humans are really good at measuring change, think of the stock market. We can change populations, you measure change. If you're in a lab, you're measuring change. So often times, we can find the function, the derivative, and then that will give us information back about the original function. What is the function that describes population? What is the function that describes the stock market? What's the function that describes this chemical equation? These are things that we got. So we go back and forth between them, and the derivative is just such a useful tool, but you have to know how to find it. This way of finding it is the limit definition of the derivative. It is the long way. And when I call it the long way, that should give you hope, because that means that there's a short way. And for those astute among us who can notice, there is a pattern. There is a pattern between 3 and the 5. I got a 15 and the -8 appears over here. So we will come up with shortcuts to compute these things. But right now, no, this is the limit definition of the derivative. This is how we find these things. And this is how these things work. Okay, so we'll Stop here and we'll let this all sink in, and then we'll do some more examples in next video.
