Introduction to derivatives
Slope and slope of a tangent line Understanding that the derivative is just the slope of a curve at a point (or the slope of the tangent line)
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- We're first exposed to the idea of a slope of a line early on
- in our algebra careers.
- But I figure it never hurts to review it a bit.
- So let me draw some axes.
- That is my y-axis.
- Maybe I should call it my f of x-axis.
- y is equal to f of x.
- Let me draw my x-axis, just like that, that is my x-axis.
- And let me draw a line, let me draw a line like this.
- And what we want to do is remind ourselves, how do we
- find the slope of that line?
- And what we do is, we take two points on the line,
- so let's say we take this point, right here.
- Let's say that that is the point x is equal to a.
- And then what would this be?
- This would be the point f of a, where the function is
- going to be some line.
- We could write f of x is going to be equal to mx + b.
- We don't know what m and b are.
- But this is all a little bit of review.
- So this is a.
- And then the y-value is what happens to the function when
- you evaluated it at a, so that's that point right there.
- And then we could take another point on this line.
- Let's say we take point b, right there.
- And then this coordinate up here is going to be
- the point b, f of b.
- Right?
- Because this is just the point when you evaluate
- the function at b.
- You put b in here, you're going to get that point right there.
- So let me just draw a little line right there.
- So that is f of b, right there.
- Actually, let me make it clear that this coordinate right here
- is the point a, f of a.
- So how do we find the slope between these 2 points, or more
- generally, of this entire line?
- Because a line has -- the slope is consistent the
- whole way through it.
- And we know that once we find the slope, that's
- actually going to be the value of this m.
- That's all a review of your algebra, but how do we do it?
- Well, a couple of ways to think about it.
- Slope is equal to rise over run.
- You might have seen that when you first learned algebra.
- Or another way of writing it, it's change in
- y over change in x.
- So let's figure out what the change in y over the change
- in x is for this particular case.
- So the change in y is equal to what?
- Well, let's just take, you can take this guy as being the
- first point, or that guy as being the first point.
- But since this guy has a larger x and a larger y,
- let's start with him.
- So, the change in y between that guy and that guy is this
- distance, right here.
- So let me draw a little triangle.
- That distance right there is a change in y.
- Or I could just transfer it to the y-axis.
- This is the change in y.
- That is your change in y, that distance.
- So what is that distance?
- It's f of b minus f of a.
- So it equals f of b minus f of a.
- That is your change in y.
- Now what is your change in x The slope is change
- in y over change in x.
- So what our change in x?
- What's this distance?
- Remember, we're taking this to be the first point,
- so we took its y minus the other point's y.
- So to be consistent, we're going to have to take this
- point's x minus this point's x.
- So this point's x-coordinate is b.
- So it's going to be b minus a.
- And just like that, if you knew the equation of this line, or
- if you had the coordinates of these 2 points, you would just
- plug them in right here and you would get your slope.
- That's straightforward.
- And that comes straight out of your Algebra 1 class.
- And let me just, just to make sure it's concrete for you,
- if this was the point 2, 3. And let's say that this, up here, was the point 5, 7.
- Then if we wanted to find the slope of
- this line, we would do 7 minus 3. That would be our change in y.
- This would be 7 and this would be 3. And then we
- do that over 5 minus 2.
- Because this would be a 5, and this would be a 2.
- And so this would be your change in x.
- 5 minus 2.
- So 7 minus 3 is 4; and 5 minus 2 is 3.
- So your slope would be 4 over 3 (4/3).
- Now let's see if we can generalize this.
- And this is what the new concept that we're going
- to be learning as we delve into calculus.
- Let's see if we can generalize this somehow to a curve.
- So let's say I have a curve.
- We have to have a curve before we can generalize
- it to a curve.
- Let me scroll down a little.
- Well, actually, I want to leave this up here, to show
- you the similarity.
- Let's say I have -- I'll keep it pretty general right now.
- Let's say I have a curve.
- I'll make it a familiar-looking curve.
- Let's say it's the curve y is equal to x squared, which
- looks something like that.
- And I want to find the slope.
- Let's say I want to find the slope at some point.
- And actually, before even talking about it, let's even
- think about what it means to find the slope of a curve.
- Here, the slope was the same the whole time, right?
- But on a curve, your slope is changing.
- And just to get an intuition for what that means --
- What's the slope over here?
- Your slope over here is the slope of the tangent line.
- The line just barely touches it.
- That's the slope over there.
- It's a negative slope.
- Then over here, your slope is still negative, but it's a
- little bit less negative.
- It goes like that.
- I don't know if I did that, drew that.
- Let me do it in a different color.
- Let me do it in purple.
- So over here, your slope is slightly less negative.
- It's a slightly less downward-sloping line.
- And then when you go over here, at the 0 point, right here,
- your slope is pretty much flat, because the horizontal line, y
- equals 0, is tangent to this curve.
- And then as you go to more positive x-es, then your
- slope starts increasing.
- I'm trying to draw a tangent line.
- And here it's increasing even more, it's increased even more.
- So your slope is changing the entire time, and this is kind
- of the big change that happens when you go from a
- line to a curve.
- A line, your slope is the same the entire time.
- You could take any two points of a line, take the change in y
- over the change in x, and you get the slope for
- the entire line.
- But as you can see already, it's going to be a little
- bit more nuanced when we do it for a curve.
- Because it depends what point we're talking about.
- We can't just say, what is the slope for this curve?
- The slope is different at every point along the curve.
- It changes.
- If we go up here, it's going to be even steeper.
- It's going to look something like that.
- So let's try a bit of an experiment.
- And I know how this experiment turns out, so it won't
- be too much of a risk.
- Let me draw better than that.
- So that is my y-axis, and that's my x-axis.
- Let's call this, we can call this y, or we can call
- this the f of x axis.
- Either way.
- And let me draw my curve again.
- And I'll just draw it in the positive coordinate, like that.
- That's my curve.
- And what if I want to find the slope right there?
- What can I do?
- Well, based on our definition of a slope, we need 2 points
- to find a slope, right?
- Here, I don't know how to find the slope with 1 point.
- So let's just call this point right here,
- that's going to be x.
- We're going to be general.
- This is going to be our point x.
- But to find our slope, according to our traditional
- algebra 1 definition of a slope, we need 2 points.
- So let's get another point in here.
- Let's just take a slightly larger version of this x.
- So let's say, let's say we have that point right there. Actually, let's do it even
- further out, just because it's going to get messy otherwise.
- So let's say we have this point right here.
- And the difference -- it's just h bigger than x.
- Or actually, instead of saying h bigger, let's just, well
- let me just say h bigger.
- So this is x + h.
- That's what that point is right there.
- So what are going to be their corresponding y-coordinates
- on the curve?
- Well, this is the curve of y is equal to f of x.
- So this point right here is going to be f of our
- particular x right here.
- And maybe to show you that I'm taking a particular x, maybe
- I'll do a little 0 here.
- This is x naught, this is x naught plus h.
- This is f of x naught.
- And then what is this going to be up here, this point up
- here, that point up here?
- Its y-coordinate is going to be f of -- f of this x-coordinate,
- which I shifted over a little bit.
- It's right there. f of this x-coordinate, which is
- f of x naught plus h.
- That's its y-coordinate.
- So what is a slope going to be between these two points that
- are relatively close to each other?
- Remember, this isn't going to be the slope
- just at this point.
- This is the slope of the line between these two points.
- And if I were to actually draw it out, it would actually be a
- secant line between, to the curve.
- So it would intersect the curve twice, once at this point,
- once at this point.
- You can't see it.
- If I blew it up a little bit, it would look
- something like this.
- This is our coordinate x naught f of x naught, and up here
- is our coordinate for this point, which would be, the
- x-coordinate would be x naught plus h, and the y-coordinate
- would be f of x naught plus h.
- Just whatever this function is, we're evaluating it at this
- x-coordinate That's all it is.
- So these are the 2 points.
- So maybe a good start is to just say, hey, what is the
- slope of this secant line?
- And just like we did in the previous example, you find the
- change in y, and you divide that by your change in x.
- Let me draw it here.
- Your change in y would be that right here, change in y, and
- then your change in x would be that right there.
- So what is the slope going to be of the secant line?
- The slope is going to be equal to, let's start with this
- point up here, just because it seems to be larger.
- So we want a change in y. so this value right here, this
- y-value, is f of x naught plus h.
- I just evaluated this guy up here.
- Looks like a fancy term, but all it means is, look.
- The slightly larger x evaluate its y-coordinate.
- Where the curve is at that value of x.
- So that is going to be, so the change in y is going to be
- a f of x naught plus h.
- That's just the y-coordinate up here.
- Minus this y-coordinate over here.
- So minus f of x naught.
- So that equals our change in y.
- And you want to divide that by your change in x.
- 242 00:11:21,09 --> 00:11:21,8 So what is this?
- This is the larger x-value.
- We started with this coordinate, so we start
- with its x-coordinate.
- So it's x naught plus h, x naught plus h.
- Minus this x-coordinate.
- Well, we just picked a general number.
- It's x naught.
- So that is over your change in x.
- Just like that.
- So this is the slope of the secant line.
- We still haven't answered what the slope is right at that
- point, but maybe this will help us get there.
- If we simplify this, so let me write it down like this.
- The slope of the secant, let me write that properly.
- The slope of the secant line is equal to the value of the
- function at this point, f of x naught plus h, minus the value
- of the function here, mine f of x naught.
- So that just tells us the change in y.
- It's the exact same definition of slope we've always used.
- Over the change in x.
- And we can simplify this.
- We have x naught plus h minus x naught.
- So x naught minus x naught cancel out, so you
- have that over h.
- So this is equal to our change in y over change in x.
- Fair enough.
- But I started off saying, I want to find the slope of
- the line at that point, at this point, right here.
- This is the zoomed-out version of it.
- So what can I do?
- Well, I defined second point here as just the first
- point plus some h.
- And we have something in our toolkit called a limit.
- This h is just a general number.
- It could be 10, it could be 2, it could be 0.02, it could be 1
- times 10 to the negative 100.
- It could be an arbitrarily small number.
- So what happens, what would happen, at least theoretically,
- if I were take the limit as h approaches 0?
- So, you know, first, maybe h is this fairly large number over
- here, and then if I take h a little bit smaller, then I'd be
- finding the slope of this secant line.
- If I took h to be even a little bit smaller, I'd be finding the
- slope of that secant line.
- If h is a little bit smaller, I'd be finding
- the slope of that line.
- So as h approaches 0, I'll be getting closer and closer to
- finding the slope of the line right at my point in question.
- Obviously, if h is a large number, my secant line is going
- to be way off from the slope at exactly that point right there.
- But if h is 0.0000001, if it's an infinitesimally small
- number, then I'm going to get pretty close.
- So what happens if I take the limit as h
- approaches zero of this?
- So the limit as h approaches 0 of my secant slope.
- Of, let me switch to green.
- f of x naught plus h minus f of x naught, that was my change in
- y, over h, which is my change in x.
- And now just to clarify something, and sometimes you'll
- see it in different calculus books, sometimes instead of an
- h, they'll write a delta x here.
- Where this second point would have been defined as x naught
- plus delta x, and then, this would have simplified to just
- delta x over there, and we'd be taking the limit as
- delta x approaches 0.
- The exact same thing.
- h, delta x, doesn't matter.
- We're taking h as the difference between one x point
- and then the higher x point, and then we're just going to
- take the limit as that approaches zero.
- We could have called that delta x just as easily.
- But I'm going to call this thing, which equals the slope
- of the tangent line, and it does equal the slope of the
- tangent line, I'm going to call this the derivative of f.
- Let me write that down.
- 319 00:15:05,58 --> 00:15:10,65 And I'm going to say that this is equal to f prime of x.
- And this is going to be another function.
- Because remember, the slope changes at every x-value.
- No matter what x-value you pick, the slope is
- going to be different.
- Doesn't have to be, but the way I drew this curve,
- it is different.
- It can be different.
- So now, you give me an x-value in here, I'll apply this
- formula over here, and then I can tell you the
- slope at that point.
- And it all seems very confusing and maybe
- abstract at this point.
- In the next video, I'll actually do an example of
- calculating a slope, and it'll make it everything a
- little bit more concrete.
Be specific, and indicate a time in the video:
At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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