# Calculus: Derivatives 1 | Taking derivatives | Differential Calculus | Khan Academy

Welcome to the presentation

on derivatives. I think you’re going to find

that this is when math starts to become a lot more fun than

it was just a few topics ago. Well let’s get started

with our derivatives. I know it sounds

very complicated. Well, in general, if I have a

straight line– let me see if I can draw a straight line

properly– if I had a straight line– that’s my coordinate

axes, which aren’t straight– this is a straight line. But when I have a straight line

like that, and I ask you to find the slope– I think you

already know how to do this– it’s just the change in y

divided by the change in x. If I wanted to find the slope–

really I mean the slope is the same, because it is a straight

line, the slope is the same across the whole line, but if I

want to find the slope at any point in this line, what I

would do is I would pick a point x– say I’d

pick this point. We’d pick a different color–

I’d take this point, I’d pick this point– it’s pretty

arbitrary, I could pick any two points, and I would figure out

what the change in y is– this is the change in y, delta y,

that’s just another way of saying change in y– and

this is the change in x. delta x. And we figured out that the

slope is defined really as change in y divided

by change in x. And another way of saying that

is delta– it’s that triangle– delta y divided by delta x. Very straightforward. Now what happens, though,

if we’re not dealing with a straight line? Let me see if I have

space to draw that. Another coordinate axes. Still pretty messy, but I

think you’ll get the point. Now let’s say, instead of just

a regular line like this, this follows the standard

y equals mx plus b. Let’s just say I had the

curve y equals x squared. Let me draw it in a

different color. So y equals x squared looks

something like this. It’s a curve, you’re probably

pretty familiar with it by now. And what I’m going to

ask you is, what is the slope of this curve? And think about that. What does it mean to take

the slope of a curve now? Well, in this line, the slope

was the same throughout the whole line. But if you look at this

curve, doesn’t the slope change, right? Here it’s almost flat, and it

gets steeper steeper steeper steeper steeper until

gets pretty steep. And if you go really far out,

it gets extremely steep. So you’re probably saying,

well, how do you figure out the slope of a curve whose

slope keeps changing? Well there is no slope

for the entire curve. For a line, there is a slope

for the entire line, because the slope never changes. But what we could try to

do is figure out what the slope is at a given point. And the slope at a given point

would be the same as the slope of a tangent line. For example– let me pick a

green– the slope at this point right here would be the same

as the slope of this line. Right? Because this line

is tangent to it. So it just touches that curve,

and at that exact point, they would have– this blue curve, y

equals x squared, would have the same slope as

this green line. But if we go to a point back

here, even though this is a really badly drawn graph,

the slope would be something like this. The tangent slope. The slope would be a negative

slope, and here it’s a positive slope, but if we took a

point here, the slope would be even more positive. So how are we going

to figure this out? How are we going to figure out

what the slope is at any point along the curve y

equals x squared? That’s where the derivative

comes into use, and now for the first time you’ll actually see

why a limit is actually a useful concept. So let me try to

redraw the curve. OK, I’ll draw my axes, that’s

the y-axis– I’ll just do it in the first quadrant– and this

is– I really have to find a better tool to do my– this is

x coordinate, and then let me draw my curve in yellow. So y equals x squared looks

something like this. I’m really concentrating

to draw this at least decently good. OK. So let’s say we want to find

the slope at this point. Let’s call this point a. At this point, x equals a. And of course this is f of a. So what we could try to do

is, we could try to find the slope of a secant line. A line between– we take

another point, say, somewhat close, to this point on the

graph, let’s say here, and if we could figure out the slope

of this line, it would be a bit of an approximation of

the slope of the curve exactly at this point. So let me draw

that secant line. Something like that. Secant line looks

something like that. And let’s say that this point

right here is a plus h, where this distance is just h, this

is a plus h, we’re just going h away from a, and then

this point right here is f of a plus h. My pen is malfunctioning. So this would be an

approximation for what the slope is at this point. And the closer that h gets,

the closer this point gets to this point, the better our

approximation is going to be, all the way to the point that

if we could actually get the slope where h equals 0, that

would actually be the slope, the instantaneous slope, at

that point in the curve. But how can we figure out what

the slope is when h equals 0? So right now, we’re saying that

the slope between these two points, it would be the

change in y, so what’s the change in y? It’s this, so that this point

right here is– the x coordinate is– my thing just

keeps messing up– the x coordinate is a plus h, and the

y coordinate is f of a plus h. And this point right here, the

coordinate is a and f of a. So if we just use the standard

slope formula, like before, we would say change in

y over change in x. Well, what’s the change in y? It’s f of a plus h– this

y coordinate minus this y coordinate– minus f of

a over the change in x. Well that change in x is this

x coordinate, a plus h, minus this x coordinate, minus a. And of course this a

and this a cancel out. So it’s f of a plus h,

minus f of a, all over h. This is just the slope

of this secant line. And if we want to get the slope

of the tangent line, we would just have to find what happens

as h gets smaller and smaller and smaller. And I think you know

where I’m going. Really, we just want to, if we

want to find the slope of this tangent line, we just have

to find the limit of this value as h approaches 0. And then, as h approaches 0,

this secant line is going to get closer and closer to the

slope of the tangent line. And then we’ll know the exact

slope at the instantaneous point along the curve. And actually, it turns out

that this is the definition of the derivative. And the derivative is nothing

more than the slope of a curve at an exact point. And this is super useful,

because for the first time, everything we’ve talked

about to this point is the slope of a line. But now we can take any

continuous curve, or most continuous curves, and find

the slope of that curve at an exact point. So now that I’ve given you the

definition of what a derivative is, and maybe hopefully a

little bit of intuition, in the next presentation I’m going to

use this definition to actually apply it to some functions,

like x squared and others, and give you some more problems. I’ll see you in the

next presentation

absolute fan of this guy!!!!!

Hkan

You are the best teacher all ever

Thank you very much

شكرا على الترجمة

My teacher doesn't teach so I do other homework in class then come to these videos for the actual help

Finally! I understood exactly what a derivative is! It's actually better a video in English than a book in Spanish.

wow that's some juicy quality

what playlist is this in? i can't seem to find it, i wanted to watch the lessons in series

This guy taught me the subject and made me understand it in 10 minutes where as my teacher took 2 hours worth of classes and made me super confused.

The best . Thank you . Revive my preliminary calculus that will hopefully leads to reviving of Fourier and Laplace transforms !!!! Back to electrical engineering motion control

My teacher is a really nice lady but I sit in class letting my mind wander but this! I totally get it now. I know how to work the formulas but never really understood why the formula is the formula. Thank God for Khans Academy and the internet.

11 years ago, this video started saving lives

SAL SOUNDS SO YOUNG

u sound so boring

İ can not understand where h came from?

Bro, your videos are good but sometimes you need to just get to the point

khan academy u helped me all throughout high school now i have to pay to watch. the world is ending

this video is very helpful thanks

Can u explain me from 8:08 to the end plz? What do u mean by as H gets smaller and smaller? What's the use of this

Shouldn’t the limit be as h approaches a

You're a doll for making these; math will be the death of me

I haven't been born and I've understood (Jk I'm 13, but age doesn't matter; we're all human beings that get crushed by the calamities of school stress :)). Do you have to know this before Calculus????

It's a really good video to understand the basic of derivation

Please Help me with this problem:- A funnel has a circular top of diameter 20cm and a height of 30cm. When the depth of the liquid in the funnel is 12cm, the liquid is dripping from the funnel at a rate of 0.2cm3s. At what rate is the depth of the liquid in the funnel decreasing at this instant?

Thank you for the explanations, I am wondering though, why do they not find the tangent at that point by placing a circle inside the curve with its circumference going through that point and making a right angle off the diameter drawn through the center from the point outside the curve? I don't understand. Is the tangent somehow specific to its curve? Thanks again.

Best explanation 👌👌👌

We need more professors like him , like jeez , been a week since we started this , and my prof can't explain the easiest part of calculus

Such an amazing way to learn,, love it!! thank u I really understand now!

I have a test tomorrow and only 2 hours to learn everything about derivates.. wish me luck 😩🙏🏼

this does not make maths more fun by the way. It is terrible

Brilliant way explanation

math not fun

pls like the vdo.

2019 still saving lives.

thanks sis for helping me :')

Plzzzz don't dislike plz

Good luck to everyone else who is going over this stuff last minute before their test.

Thank you ☺️🧐

oh yeah yeah

ty u so much

240p oh yeah

7:27 thank you my OCD was going through the roof

I won't complain,only provide an good channel for learning calculus and math in general,

#3blue1brown

The money shot is at 7:55

This video seems old compared to other khan videos. The visual and audio quality isn’t as great, but it’s still a good explanation.

My life is saved, lol.

0:06 you already lost me dog

this was uploaded 12 years ago and still just as useful

Thank you so much you saved my Life literally !!

Anyone still watching in 2019? #thanksSal

AWESOME LOVE KHAN ACADEMY

why am i spending so much on my college course when i can just watch this for free lol

10 mins of Khan > 75 minute calculus lecture

WHY IS EVERYONE ELSE SO COMPLICATED ABOUT IT WHEN U EXPLAINED IT SO SIMPLY IN A 10 MINUTE VIDEO. THATS WAY LESS THAN MY LECTURE BUT WHAT DID I GAIN FROM LECTURE? NOTHING. JUST COMPLICATIONS

Thank you

I've made 98 average in advanced Calculus. Now 7 years later I've forgotten everything I know and you saved my life lol good job man. Thanks.

Use paint in the computer application to make slopes instead of just drawing it out. It'll save you some trouble. Lol if anyone remembers the old school paint that Microsoft has in every computer. That's the one.

5:16 he got the rumblies in the tumblies

Hey khan academy, get a new mic please.

2019?

LEGEND!

see-cond?

THERE'S HOPE FOR MANKIND … I'm so proud of you

Bro

Well he taught nicely but why using computer to teach…i mean it takes alot of time to draw

Should I stop paying my teacher now?

What your application to make this content??

Why is it that I understand integrals more than derivatives??? ;_;

Black panther to cap :

Give this man a shield 🛡.

Me to khan Academy:

Give this man a better software to teach

Excellent tutorial! Thank you so much for sharing and explaining the complex concept nicely!!!

Getting too hung up with the whiteboard technology

This is the point of my life when i finally realized that i am one of the stupid ones.

Thanks Khan Academy!! Grettings from Argentina

Sir, which software u use to interpret 3d graphs…example z= f(x) = x2 + y2

Great instruction unless very poor tools for presentation, it need a pen mouse and PowerPoint inherited widgets to create awesome shapes and graphs

honestly, thank you so much. I have a test on derivatives tomorrow and I’ve completely forgotten how to do everything

anyone at 2019. khan acdemy still saving the students

I want to find the software that teacher used to draw and write in the video. Many thanks

Aw, how excited he was about maths back then

"Darn straight" 😀

Legends say watch this video in 1.25x

its my first year in aeronautical engineering, thank you 😊

This is by far still the best definition of derivatives I can find on youtube. Other youtubers don't even explain that it's the slope that you're looking for.

I’m not even in highschool yet and you seem to explain it purely, without any strenous effort. Just how do you do it? How?

Listen please, similarly My math song f'(a) in my Youtube channel.

Who's here in 2019 screwing with their lives?

This is thee most beautifully simplistic intro to derivatives. Summer Calc class at UC Davis, and the professor has clearly chosen the wrong occupation. Thank you Kahn for your talent and hard work.

Is there anyway I can find the name of the speaker so I can send him a personal email of me thanking him? Khan Academy is literally going to get me through this AP Calc class…

Donna Starr sent me here.

❤️❤️❤️

Fail