Big Picture ：Derivatives

PROFESSOR: OK, hi.This is the second in my videos about the main ideas,the big picture of calculus.And this is an important one, because I want to introduceand compute some derivatives.

And you'll remember the overall situation is we havepairs of functions, distance and speed, function 1 andfunction 2, height of a graph, slope of the graph, height ofa mountain, slope of a mountain.

And it's the connection between those two thatcalculus is about.

Relationship Between Function

And so our problem today is, you could imagine we have anairplane climbing.Its height is y as it covers a distance x.And its flight recorder will--Well, probably it has two flight recorders.Let's suppose it has.Or your car has two recorders.

One records the distance, the height, the total amountachieved up to that moment, up to that time, t,or that point, x.

The second recorder would tell you at every instantwhat the speed is.So it would tell you the speed at all times.

Do you see the difference?The speed is like what's happening at an instant.The distance or the height, y, is the total accumulation ofhow far you've gone, how high you've gone.

And now I'm going to suppose that this speed, this secondfunction, the recorder is lost. But the information isthere, and how to recover it.So that's the question.How, if I have a total record, say of height--I'll say mostly with y of x.I write these two so that you realize that letters are notwhat calculus is about.It's ideas.

And here is a central idea.if I know the height--as I go along, I know the height, it could go down--how can I recover from that height what the slope is ateach point?

So here's something rather important, that's the notationthat Leibniz created, and it was a good, good idea for thederivative.And you'll see where it comes from.

But somehow I'm dividing distance up by distanceacross, and that ratio of up to across is a slope.

Slope of the Great Functions of Calculus

So let me develop what we're doing.So the one thing we can do and now will do is, for the greatfunctions of calculus, a few very special,very important functions.We will actually figure out what the slope is.

These are given by formulas, and I'll find a formula forthe slope, dy/dx equals.And I won't write it in yet.Let me keep a little suspense.But this short list of the great functions istremendously valuable.The process that we go through takes a little time, but oncewe do it it's done.Once we write in the answer here, we know it.

And the point is that other functions of science, ofengineering, of economics, of life, come from thesefunctions by multiplying--I could multiply that times that--and then I need a product rule, the rule for thederivative, the slope of a product.I could divide that by that.So I need a quotient rule.

I could do a chain.And you'll see that's maybe the best and most valuable.e to the sine x, so I'm putting e to the x togetherwith sine x in a chain of functions, e to the sine of x.Then we need a chain rule.That's all coming.

y=x^n

Let me start--Well, let me even start by giving away the main facts forthese three examples, because they're three you want toremember, and might as well start now.The x to the nth, so that's something if n is positive.x to the nth climbs up.

Let me draw a graph here of y equals x squared, becausethat's one we'll work out in detail, y equals x squared.

So this direction is x.This direction is y.And I want to know the slope.And the point is that that slope is changing as I go up.So the slope depends on x.The slope will be different here.So it's getting steeper and steeper.I'll figure out that slope.

For this example, x squared went inas 2, and it's climbing.If n was minus 2, just because that's also on our list, ncould be negative, the function would be dropping.You remember x to the minus 2, that negative exponent meansdivide by x squared.x to the minus 2 is 1 divided by x squared,and it'll be dropping.So n could be positive or negative here.

So I tell you the derivative.

The derivative is easy to remember, the set number n.It's another power of x, and that power isone less, one down.You lose one power.

I'm going to go through the steps here for n is equal to2, so I hope to get the answer 2 times x to the 2 minus 1will just be 1, 2x.

y=sinx

But what does the slope mean?That's what this lecture is really telling you.

I'll tell you the answer for if it's sine x going in,beautifully, the derivative of sine x is cos x, the cosine.The derivative of the sine curve is the cosine curve.

You couldn't hope for more than that.And then we'll also, at the same time, find the derivativeof the cosine curve, which is minus the sine curve.It turns out a minus sine comes in because the cosinecurve drops at the start.

y=e^x

And would you like to know this one?e to the x, which I will introduce in a lecture comingvery soon, because it's the function of calculus.And the reason it's so terrific is, the connectionbetween the function, whatever it is, whatever this number eis to whatever the xth power means, the slope is the sameas the function, e to the x.

That's amazing.As the function changes, the slope changesand they stay equal.

Slope

Really, my help is just to say, if you know those three,you're really off to a good start, plus therules that you need.All right, now I'll tackle this particular one and say.

y=x^2

what does slope mean?So I'm given the recorder that I have. Thisis function 1 here.This is function 1, the one I know.And I know it at every point.

If I only had the trip meter after an hour or two hours orthree hours, well, calculus can't do the impossible.

It can't know, if I only knew the distance reached aftereach hour, I couldn't tell what it did over that hour,how often you had to break, how often you accelerated.

I could only find an average speed over that hour.That would be easy.So averages don't need calculus.

It's instant stuff, what happens at a moment, what isthat speedometer reading at the moment x, say, x equal 1.

What is the slope?Yeah, let me put in x equals 1 on this graph and x equals 2.And now x squared is going to be at height 1.If x is 1, then x squared is also 1.If x is 2, x squared will be 4.

What's the average?Let me just come back one second tothat average business.The average slope there would be, in a distance across of 1,I went up by how much?3.I went up from 1 to 4.I have to do a subtraction.Differences, derivatives, that's the connection here.

So it's 4 minus 1.That is 3.So I would say the average is 3/1.But that's not calculus.Calculus is looking at the instant thing.

And let me begin at this instant, at that point.

What does the slope look like to you at that point?At at x equals 0, here's x equals 0, andhere's y equals 0.We're very much 0.You see it's climbing, but at that moment, it's like it juststarted from a red light, whatever.

The speed is 0 at that point.And I want to say the slope is 0.That's flat right there.That's flat.

If I continued the curve, if I continued the x squared curvefor x negative, it would be the same as for x positive.Well, it doesn't look very the same.Let me improve that.It would start up the same way and be completely symmetric.Everybody sees that, at that 0 position, thecurve has hit bottom.

Actually, this will be a major, majorapplication of calculus.You identify the bottom of a curve by the fact that theslope is 0.It's not going up.It's not going down.It's 0 at that point.

slope at a point

But now, what do I mean by slope at a point?

Here comes the new idea.If I go way over to 1, that's too much.I just want to stay near this point.I'll go over a little bit, and I call thatlittle bit delta x.So that letter, delta, signals to ourminds small, some small.And actually, probably smaller than I drew it there.And then, so what's the average?I'd like to just find the averagespeed, or average slope.

If I go over by delta x, and then how high do I go up?Well, the curve is y equals x squared.So how high is this?So the average is up first, divided by across.Across is our usual delta x.How far did it go up?

Well, if our curve is x squared and I'm at the point,delta x, then it's delta x squared.That's the average over the first piece, over short, overthe first piece of the curve.Out is-- from here out to delta x.OK.

Now, again, that's still only an average, because delta xmight have been short.I want to push it to 0.That's where calculus comes in, taking the limit ofshorter and shorter and shorter pieces in order tozoom in on that instant, that moment, that spot where we'relooking at the slope, and where we're expecting theanswer is 0, in this case.

And you see that the average, it happens tobe especially simple.Delta x squared over delta x is just delta x.

So the average slope is extremely small.And I'll just complete that thought.So the instant slope-- instant slope at 0, at x equals 0, Ilet this delta x get smaller and smaller.I get the answer is 0, which is just what I expected it.And you could say, well, not too exciting.But it was an easy one to do.It was the first time that we actually went through thesteps of computing.

This is a, like, a delta y.This is the delta x.Instead of 3/1, starting here I had delta xsquared over delta x.That was easy to see.It was delta x.And if I move in closer, that average slope is smaller andsmaller, and the slope at that instant is 0.No problem.

The travel, the climbing began from rest, butit picked up speed.The slope here is certainly not 0.We'll find that slope.We need now to find the slope at every point.OK.That's a good start.

Now I'm ready to find the slope at any point.Instead of just x equals 0, which we've now done, I betterdraw a new graph of the same picture, climbing up.Now I'm putting in a little climb at some point x here.I'm up at a height, x squared.I'm at that point on the climb.I'd like to know the slope there, at that point.

How am I going to do it?

I will follow this as the central dogma of calculus, ofdifferential calculus, function 1 to function 2.Take a little delta x, go as far as x plus delta x.That will take you to a higher point on the curve.That's now the point x plus delta x squared, because ourcurve is still y equals x squared in thisnice, simple parabola.OK.

So now I you look at distance across and distance up.So delta y is the change up.Delta x is the across.And I have to put what is delta y.

I have to write in, what is delta y?It's this distance up.It's x plus delta x squared.That's this height minus this height.I'm not counting this bit, of course.It's that that I want.That's the delta y, is this piece.

So it's up to this, subtract x squared.That's delta y.That's important.Now I divide by delta x.This is all algebra now.Calculus is going to come in a moment, but not yet.

For algebra, what do I do?I multiply this out.I see that thing squared.I remember that x is squared.And then I have this times this twice.2x delta x's, and then I have delta x squared.And then I'm subtracting x squared.So that's delta y written out in full glory.

I wrote it out because now I can simplify bycanceling the x squared.I'm not surprised.Now, in this case, I can actually do the division.Delta x is just there.So it leaves me with a 2x.Delta x over delta x is 1.And then here's a delta x squared over a delta x, sothat leaves me with one delta x.As you get the hang of calculus, you see theimportant things is like this first order, delta x to thefirst power.Delta x squared, that, when divided by delta x, gives usthis, which is going to disappear.That's the point.

This was the average over a short but still not instantrange, distance.Now, what happens?Now dy/dx.So if this is short, short over short, this is darn shortover darn short.That d is, well, it's too short to see.So I don't actually now try to separate a distance dy.This isn't a true division, because it's effectively 0/0.And you might say, well, 0/0, what's the meaning?Well, the meaning of 0/0, in this situation, is, I take thelimit of this one, which does have a meaning, because thoseare true numbers.They're little numbers but they're numbers.

And this was this, so now here's the big step, leavingalgebra behind, going to calculus in order to getwhat's happening at a point.I let delta x go to 0.And what is that?So delta y over delta x is this.What is the dy/dx?So in the limit, ah, it's not hard.Here's the 2x.It's there.Here's the delta x.In the limit, it disappears.

So the conclusion is that the derivative is 2x.So that's function two.That's function two here.That's the slope function.That's the speed function.Maybe I should draw it.Can I draw it above and then I'll put the board back up?

So here's a picture of function 2, the derivative, orthe slope, which I was calling s.So that's the s function, against x.x is still the thing that's varying, or it could be t, orit could be whatever letter we've got.And the answer was 2x for this function.

So if I graph it, it starts at 0, and it climbssteadily with slope 2.So that's a graph of s of x.And for example--yeah, so take a couple of points on that graph--at x equals 0, the slope is 0.And we did that first. And we actually got it right.The slope is 0 at the start, at the bottom of the curve.

At some other point on the curve, what's the slope here?Ha, yeah, tell me the slope there.At that point on the curve, an average slope was 3/1, butthat was the slope of this, like, you know--sometimes called a chord.That's over a big jump of 1.Then I did it over a small jump of delta x, and then Ilet delta x go to 0, so it was an instant infinitesimal jump.

So the actual slope, the way to visualize it is that it'smore like that.That's the line that's really giving theslope at that point.That's my best picture.It's not Rembrandt, but it's got it.And what is the slope at that point?Well, that's what our calculation was.

It found the slope at that point.And at the particular point, x equals 1, the height was 2.The slope is 2.The actual tangent line is only-- is there.You see?It's up.Oh, wait a minute.Yeah, well, the slope is 2.I don't know.This goes up to 3.It's not Rembrandt, but the math is OK.

So what have we done?We've taken the first small step and literally I could saysmall step, almost a play on words because that's thepoint, the step is so small--to getting these great functions.

y = sinx

Before I close this lecture, can I draw this pair, function1 and function 2, and just see that the movement of thecurves is what we would expect.

So let me, just for one more good example, great example,actually, is let me draw.Here goes x.In fact, maybe I already drew in the first letter, lecturethat bit out to 90 degrees.Only if we want a nice formula, we better call thatpi over 2 radians.And here's a graph of sine x.This is y.This is the function 1, sine x.

And what's function 2?What can we see about function 2?Again, x.We see a slope.This is not the same as x squared.This starts with a definite slope.And it turns out this will be one of the most importantlimits we'll find.

We'll discover that the first little delta x, which goes upby sine of delta x, has a slope that gets closer andcloser to 1.Good.Luckily, cosine does start at 1, so we're OK so far.Now the slope is dropping.

And what's the slope at the top of the sine curve?It's a maximum.But we identify that by the fact that the slope is 0,because we know the thing is going to go down here and gosomewhere else.The slope there is 0.The tangent line is horizontal.And that is that point.It passes through 0.The slope is dropping.So this is the slope curve.

And the great thing is that it's the cosine of x.And what I'm doing now is not proving this fact.I'm not doing my delta x's.That's the job I have to do once, and it won't be today,but I only have to do it once.But today, I'm just saying it makes sensethe slope is dropping.In that first part, I'm going up, so the slope is positivebut the slope is dropping.

And then, at this point, it hits 0.And that's this point.And then the slope turns negative.I'm falling.So the slope goes negative, and actuallyit follows the cosine.So I go along here to that point, and then I can continueon to this point where it bottoms out againand then starts up.

So where is that on this curve?Well, I'd better draw a little further out.This bottom here would be the--This is our 2π​.This is our π, 180 degrees, everybody would say.So what's happening on that curve?The function is dropping, and actually it's dropping itsfastest. It's dropping its fastest at that point, whichis the slope is minus 1.

And then the slope is still negative, but it's not sonegative, and it comes back up to 0 at 23π​ So this is thepoint, 23π​.And this has come back to 0 at that point.And then it finishes the whole thing at 2π.This finishes up here back at 1 again.It's climbing.

Conculsions

All right, climbing, dropping, faster, slower, maximum,minimum, those are the words that make derivativesimportant and useful to learn.And we've done, in detail, the first of ourgreat list of functions.Thanks.

FEMALE SPEAKER: This has been a production of MITOpenCourseWare and Gilbert Strang.Funding for this video was provided by the LordFoundation.To help OCW