# 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 introduce and compute some derivatives. And you’ll remember the overall

situation is we have pairs of functions, distance

and speed, function 1 and function 2, height of a graph,

slope of the graph, height of a mountain, slope

of a mountain. And it’s the connection

between those two that calculus is about. And so our problem today is, you

could imagine we have an airplane 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 amount achieved up to that moment,

up to that time, t, or that point, x. The second recorder would tell

you at every instant what 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 of how far you’ve gone, how

high you’ve gone. And now I’m going to suppose

that this speed, this second function, the recorder is lost.

But the information is there, 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 not what 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 at each point? So here’s something rather

important, that’s the notation that Leibniz created, and it was

a good, good idea for the derivative. And you’ll see where

it comes from. But somehow I’m dividing

distance up by distance across, and that ratio of

up to across is a slope. So let me develop what

we’re doing. So the one thing we can do and

now will do is, for the great functions 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 for the 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 is tremendously valuable. The process that we go through

takes a little time, but once we do it it’s done. Once we write in the answer

here, we know it. And the point is that other

functions of science, of engineering, of economics,

of life, come from these functions by multiplying– I could multiply that

times that– and then I need a product

rule, the rule for the derivative, 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 together with sine x in a chain of

functions, e to the sine of x. Then we need a chain rule. That’s all coming. Let me start– Well, let me even start by

giving away the main facts for these three examples, because

they’re three you want to remember, 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, because that’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 in as 2, and it’s climbing. If n was minus 2, just because

that’s also on our list, n could be negative, the function

would be dropping. You remember x to the minus 2,

that negative exponent means divide 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 is one less, one down. You lose one power. I’m going to go through the

steps here for n is equal to 2, so I hope to get the answer

2 times x to the 2 minus 1 will just be 1, 2x. 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 derivative of the cosine curve, which

is minus the sine curve. It turns out a minus sine comes

in because the cosine curve drops at the start. And would you like

to know this one? e to the x, which I will

introduce in a lecture coming very soon, because it’s the

function of calculus. And the reason it’s so terrific

is, the connection between the function, whatever

it is, whatever this number e is to whatever the xth power

means, the slope is the same as the function, e to the x. That’s amazing. As the function changes,

the slope changes and they stay equal. Really, my help is just to say,

if you know those three, you’re really off to a

good start, plus the rules that you need. All right, now I’ll tackle this

particular one and say, what does slope mean? So I’m given the recorder

that I have. This is 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 or three hours, well, calculus

can’t do the impossible. It can’t know, if I only knew

the distance reached after each 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 is that 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 to that 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, and here’s y equals 0. We’re very much 0. You see it’s climbing, but at

that moment, it’s like it just started 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 curve for 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, the curve has hit bottom. Actually, this will

be a major, major application of calculus. You identify the bottom of a

curve by the fact that the slope is 0. It’s not going up. It’s not going down. It’s 0 at that 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 that little bit delta x. So that letter, delta,

signals to our minds 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 average speed, 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, over the first piece of the curve. Out is– from here

out to delta x. OK. Now, again, that’s still only

an average, because delta x might have been short. I want to push it to 0. That’s where calculus comes

in, taking the limit of shorter and shorter and shorter

pieces in order to zoom in on that instant, that

moment, that spot where we’re looking at the slope, and where

we’re expecting the answer is 0, in this case. And you see that the average,

it happens to be 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, I let 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 the steps of computing. This is a, like, a delta y. This is the delta x. Instead of 3/1, starting

here I had delta x squared over delta x. That was easy to see. It was delta x. And if I move in closer, that

average slope is smaller and smaller, and the slope

at that instant is 0. No problem. The travel, the climbing

began from rest, but it 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 better draw 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, of differential 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 our curve is still y equals

x squared in this nice, 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 by canceling 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, so that leaves me with

one delta x. As you get the hang of

calculus, you see the important things is like this

first order, delta x to the first power. Delta x squared, that, when

divided by delta x, gives us this, which is going

to disappear. That’s the point. This was the average over a

short but still not instant range, distance. Now, what happens? Now dy/dx. So if this is short, short over

short, this is darn short over 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 the limit of this one, which does

have a meaning, because those are true numbers. They’re little numbers

but they’re numbers. And this was this, so now here’s

the big step, leaving algebra behind, going to

calculus in order to get what’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, or the 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, or it 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 climbs steadily 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, but that 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 I let delta x go to 0, so it was

an instant infinitesimal jump. So the actual slope, the way to

visualize it is that it’s more like that. That’s the line that’s

really giving the slope 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 say small step, almost a play on

words because that’s the point, the step is so small– to getting these great

functions. Before I close this lecture, can

I draw this pair, function 1 and function 2, and just see

that the movement of the curves 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, lecture that bit out to 90 degrees. Only if we want a nice formula,

we better call that pi 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 important limits we’ll find. We’ll discover that the first

little delta x, which goes up by sine of delta x, has a slope

that gets closer and closer 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 go somewhere 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 sense the slope is dropping. In that first part, I’m going

up, so the slope is positive but 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 actually it follows the cosine. So I go along here to that

point, and then I can continue on to this point where

it bottoms out again and 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 pi/2. This is our pi, 180 degrees,

everybody would say. So what’s happening

on that curve? The function is dropping, and

actually it’s dropping its fastest. It’s dropping its

fastest at that point, which is the slope is minus 1. And then the slope is still

negative, but it’s not so negative, and it comes back up

to 0 at 3 pi/2 So this is the point, 3 pi/2. And this has come back

to 0 at that point. And then it finishes the

whole thing at 2 pi. This finishes up here

back at 1 again. It’s climbing. All right, climbing, dropping,

faster, slower, maximum, minimum, those are the words

that make derivatives important and useful to learn. And we’ve done, in detail,

the first of our great list of functions. Thanks. FEMALE SPEAKER: This has been

a production of MIT OpenCourseWare and

Gilbert Strang. Funding for this video was

provided by the Lord Foundation. To help OCW continue to provide

free and open access to MIT courses, please

make a donation at ocw.mit.edu/donate.

Quote" is not rembrandt, but the math is ok". lol Deffo not a Rembrandt…But the math is brilliant and so are you, as a teacher , Mr Strang !

Thank you so very much Dr Strang! This is really helping me move forward in my college calc prereq! I appreciate the work you put into every video!

Mr. Strang is a Professor of Mathematics. He has spent a great deal of effort and time working toward a PhD; meaning he's earned the title of, "Dr. Strang". It might seem petty to address him with the proper title, but it's kind of a big deal.

You are right Sir my Bad. No insult intended.

I enjoyed it (and learned) very much. Tks.

so nice.

This is very helpful for my first year engineering course..

Gracias profesor por su enseñanza y gracias a mitopencourseware por su apoyo a la divulgacion del conocimiento para todos los habitantes de nuestro planeta Tierra. El conocimiento de las matematicas nos ayuda a proteger nuestro planeta y a dar bienestar a la gente.

I don't want to do my philosophy paper so bad that I'm here studying Calculus haha

This is helpful, thanks for the upload!!

very informative, thanks

Dr. Gilbert Strang is na outstanding example of what teaching means. Greetings from Brazil.

I remember this stuff at high school but was completely boring then. I love it now. Feels like Im peering through a keyhole at the secrets of the cosmos.

He reminds me of my old physics teacher whom (I have no idea if I just used whom correctly hah) I had in, oh well, 7th grade or so, and he was the one introducing us to atoms and molecules and all that stuff. Everyone thought he was boring and dull, but I think he was a nice guy – just like this guy (: Respect for that old man. And thanks for a great lesson! I've always dreamt of how it was to be at MIT or Harvard or Oxford or anything like that!

Nice work professor.

Just what the doctor ordered. I wanted to delve into my old textbooks, but had lost a lot of the ideas that made sense of them. This was the perfect refresher to get started. Thank you very much!

Thank you Dr. Gilbert Strang. Really a wonderful lecture

Well done sir!

Its funny how i may be the only business student who has to go through calculus, and not having this in high school just makes it even more harder for me to grasp the basic concepts 🙁

Learned more about calculus in these 30 minutes than in my entire schooling. Dr. Gilbert you are the best. Hats off to you.

We are spoiled. I am going back to grad school so I am brushing up on Calculus and let me say the second time around has been cake (in terms of learning and accessibility of information). I can't believe the difference in the level of exposure to information between then (late 90s) and now.

Back when I was taking Calculus I never thought that someday I would be listening to and watching Calculus lectures from a professor at MIT in my house or on a walk. What a wonderful time to learn. This program at MIT is truly exceptional.

Even my teens are watching along and getting exposure. Strang is perfect.

Thanks so much. This is of such a great help!

God bless you sir.

[email protected]

Not only i can understand calculus but i can think it

Finally someone who gives us the WHY behind math and "goes under the hood" as Keith Devlin says. Excellent series on calculus

Thank you Dr.Gilbert and to the MIT people behind this initiative. This is a perfect example of how much difference an excellent teacher can bring about in the learning process and make seemingly difficult concepts graspable to anyone at all. I have started appreciating calculus so much more.

His manners are charming and his attempt to make things really clear from the very beginning quite successful. 14:30: the formula of average change is wrong because he writes (Delta x) ^2 over Delta x instead of Delta x^2 over Delta x; he goes further to equate the quotient with Delta x, which is 1 instead of the 3 he had previously computed.

I like that ! Thank you for uploading. Love you.

Recommend watching at 1.5X speed if you are following along well. It helps save time while getting the full information. Feel free to rewind or pause if you need a minute to process something.

MIT is Amassing woooooooooooooow

This is the reason why MIT is a great school to go to. They teach conceptual understanding and ideas instead of a step by step method to solving an equation. It's annoying that teachers dumb down calculus simply as a means to prepare for a test or an exam. This is why we're only as strong as our weakest link. Teachers shouldn't be judged by their flunk rates. Those who want to succeed, will.

Bom profesor

Professor awesome explains what its calculus — not just solve equations. He explains the reasoning behind the madness. Well done!

thanks for sharing such wonderful knoledge

Watch at 2x speed helps you get through them faster and makes sure your pay attention

In 2x + Delta x . How Delta x is ignored in new function ?

With a professor like Mr. Strang, there is no need to ask a bunch of questions in the end. Well explained! …and worth the 30 minutes.

Good lecture! Shame though i wanted to see the sin x proof 🙁

haha "we have taken a small step – literally and actually that is the point of calculus" 🙂

best best explaination…..

I see… but how do I use it in my every day life? 😀

I`ll call it the beauty of making sence from conceptual ideas .

Just great explanation

excellent

really informational and had a nice experience learning calculus with prof. Gilbert Strang

I wish you were my teacher. That, or some old bearded man that is as competent as you.

I like explaining it like this

if f(x) = x^2 then f(10)=100 and f(11)=121 so its up +21

this thing caller a "derivative" lets us estimate the change

d/dx f(x) = 2x which we also write as f'(x)

now f'(x) evaluated at 10 is f'(10) = 20 which is pretty close to 21 (5% err)

it works better with bigger numbers

f(100) = 10000 f(101) = 10201 so its up +201

f'(100) = 200 which is pretty close to 201 (0.5% err)

and it also works better with smaller changes

f(10) = 100 f(10.1)= 102.01 so its up +2.01

f'(10) = 20 so it should go up 20 for each unit of input increase

so 20 * 0.1 = 2 which is pretty close to 2.01

if f(x) = x^3 …….

Excellent sir, Thanks for the help. Concept cleared, Hope in the future I get a teacher just like you.

at 24:30 when he was little confused and told that slop is equal to 2 and then he moves to function(1) and from the instantaneous point means x=1 from where he wanted to calculate the slope he added 2+1 and seems like the tangent line shows that slop at graph on function(1) should be at that point means 3….am i right here? is this what he wanted to tell?

so precise! cutting through whole chapters in just 30 minutes. wow.

My dream is to find a girl who is as hot as i am handsome and we can sit around at night watching university lectures in mathematics for fun. Happy Valentines day.

Everything makes Sense. Literally Goosebumps moment. Thanks a ton. Free education empowers you exceedingly. Thanks to MIT and the sponsor. You have impacted many lives, I believe.

24:40 A small step for man, but a huge leap for Calculus!)

Thank God I finally got an explanation as to why derivatives work. I missed a day of math for a wrestling tournament and never got it explained to me, and now that year and a half long itch has been starched.

15 dislikes were by Newton who is jealous of Dr. Strang's awesome ability to teach calculus at MIT!

We're such a lucky generation, i'd think of only 20 years back how it would be hard to reach such understanding.

Big thanks for Prof. Strang for how unique way for teaching the meaning and not the steps !!!!!!!!!

And For MIT open courseWare: you are making the world better 🙂

great lectute

he took the limit when tend to what value?

Thanks for uploading this gold! I'm watching this for my 11th grade exams

I thought this was going to be about economics.

Gilbert is the coolest guy ever.

ulan şöyle tane tane anlatan olmadı bize bea

Thankyou very much prof strang…

21:10 I get that 2x + ∆x = 2x if ∆x is very nearly zero. However, how was it possible to get to 2x + ∆x in the first place if ∆x = 0? You'd be dividing by zero, no? I'd be grateful if anybody could explain this.

It's not rambrandt, but the math is O. K. O.K, professor, who can argue?

A professional professor + Chalkboard + Jumbo chalk + classic approach in teaching = MIT is fortunate.

Man, if only I looked for MIT open courses before. I would have brushed up on these concepts 5 years ago. Still not late to learn from great professors.

Omg, this is the clearest explanation of derivatives ever!! Thank you for the video!

Excellent

calculas

He's so good to listen to. He sounds like a narrator in a documentary. So entertaining!

"Letters are not what Calculus is about. It's ideas." – Gilbert Strang

thank you ……very interesting

this "explanation" seems all over the place, even in my high school the argument was clearer

,, and A?C power follows the "y=sin x" curve/line.

What's the average slope and why do we need it? Doesn't taking the limit, aka going dy/dx mean we're again taking an average slope at an infinitely small difference between points, say 3 and 3.00…001, instead of taking an average delta/delta slope between 3 and 4? So the slope is the rate of change, for every x we get this much y, so an average slope would be, between points a and b, no matter how far they are apart, in average, for this x we get this much y. So the slope at a point does exactly the same thing, it approximates that around the point ex. 3, for this much x we get this much y, so that's the rate of change around point 3, but since we try to get infinitely small movements around the point 3, we might as well say that that's the rate of change "exactly" at 3. Wait so the whole point of differential calculus is just to see the rate of change at a certain point? I can't see the big picture…

Excellent refresher lecture after many years. Plus, I was excited to learn that the "d" in dy/dx stands for "darn" .

Thanks Professor Gilbert Strang 🙂 This videos is really helpful and fun to watch

At first i used to just see x's as x but now i see a whole different meaning to the terms functions, slope etc, thank you professor

Cant beleive how clearly he has explained it ( in such a humble manner ) , great human & gift to us

Just look at the passion in Prof. Strang’s explanations !! Amazing …

the most loaded 6minutes on youtube ever…thank you

Read up on infinitesimals. It’s a fascinating area of mathematicsand the history of calculus as we know it today.

So is Function 2 the derivative of function 1?

rate of change is the derivative of height?

This is the best mathetics class I ever had! Thank you professor!

I would have never been math phobic if only I had teachers like Prof.Strang in my school days.

very helpful ,thks

JuliaCon 2018 | A conversation with Gilbert Strang

https://www.youtube.com/watch?v=gGYcSjrqbjc

Professor Strang is still teaching.

—————————————————————————————————————–

18.085/0851 – Computational Science And Engineering I (Fall 2018)

Instructor: Gil Strang

This is the best mathetics class I ever had! Thank you professor!

It felt a loss of time to me. This level of calculus needs only a decent book to read. It would be unfair to comment on the lecturer from only one lecture, but I didn't like it.

slope and speed are the first prime or the first derivative of a given function. If the car is travelling at y=mx+b, then by applying the power rule of a derivative then it's speed is m. In this discussion, Professor Strang clearly and wonderfully shown to us the concept/idea behind the power rule more than just as nX^n-1. Thank you Professor Gilbert Strang.

This is an old video and I dont thinl I will receive an answer but how a slope could help you get the derivative? I get lost here

Professor Strang is the Paul Harvey of higher mathematics.

this Gilbert guy rocks, holy shit this guy is amazing

Beautiful lecture

https://youtu.be/z_WsUNs6-iw

MIT professor is greater than Oxford professor

I thought the purpose of derivatives was to try to make fractions seem larger than they are in reality LOL

👍👍👍👍👍

Nice