Calculating slope of tangent line using derivative definition | Differential Calculus | Khan Academy
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- Опубліковано 9 лют 2025
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Differential calculus on Khan Academy: Limit introduction, squeeze theorem, and epsilon-delta definition of limits.
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My teacher is so shit when I watch his videos it feels like I'm learning things for the first time. You're the goat thank you
same
The greatness of this video is in its smooth and logical explanation of the difference between a slope of a secant line and a slope of a tangent! Thank you very much, Sal!!!
R8
It was my 2nd lecture and I'm amazed by the way it was explained.
God bless this wonderful thing called Khan academy. In 10 minutes I can learn what it took Isaac Newton(I guess) years to figure out.
Sal you are too awesome, after watching your videos I get the distinct impression that my lecturers are coaching me for exams to boost the pass rate as opposed to making sure that I understand the concepts.
Calculus is actually super intuitive, who would have thought!
Just as usual, Sal is the master at explaining complex ideas in simple ways.
For some of those who may get confused with the division of ⌂y/⌂x at 6:27, another way to do the division would be to factor out a ⌂x from the top. The original equation was (6⌂x+(⌂x)²)/⌂x, we can rewrite the numerator as ⌂x(6+⌂x), leaving us with the fraction (⌂x(6+⌂x))/⌂x. We can divide the ⌂x we factored out from 6⌂x+(⌂x)² with the ⌂x on the bottom, which leaves us with the same result: 6+⌂x. Rewriting things with terms factored out is a good way to avoid "hairy math" and careless mistakes.
There should be a whole semester of calculus in UA-cam...... pfff classrooms?? forget about it.
I just need to watch this guy!
As a testimonial to your awesomeness, i went from an average of 50-75% in math to a whopping 99% in my derivatives test. All thanks to these great vids! 99%! Im still in shock.
You're so reliable, and in my opinion, books need you too. I have this Pre-Calculus book, and it uses the EXACT same example as you gave, and by the way, it was published 10 years later...so it could've copied you...
at 6:26 I hope this clears up any confusion about the 6 + delta x.
I was having a tough time with it myself wondering why it wasn't 6 + delta x^2
The reason it is 6 + delta x is because the upper portion should be in parentheses like this:
(6dx + dx^2) / dx
now what you need to do is factor out dx from the top giving you
dx(6+dx) / dx
cancel out the delta xs and you are left with 6 + dx
If you cant see it, try just using x instead of dx for the sake of clarity
(6x+x^2)/x = x(6+x)/x = 6 + x.
Thank you
Thanks man, I totally forgot about this comment. One year on and I got the Mathematics award for my whole school. I think he just inspired me to be passionate about maths
great work, you dont know what a difference you are making to this world. you have pioneered a new form of learning in a much easier and much more understandable way. you sal should be knighted for your time and effort you are giving to the world. you are a generous man who gives up your valueable time to help people whom you dont even know. great work again. from all youtube fans .
Amazing! everything is so clearly explained. I'm definitely going to be looking at all of your calculus videos! Thank you soooo much!
Sal, you never know how much joy you bring to all students all over the world! (including me) ^^
Generally the ray of knowledge is a tangent of my brain, thanks for turning it into a secant.
Man this is awesome, so easy to understand :D this guy should get paid
So the formula that he derived for the slope of the secant line was f(x + dx) - f(x) / dx. In this particular problem, he said that x would be 3. Since f(x) is x^2, we can plug in the function and variables. So the formula becomes (3 + dx)^2 - (3^2) /dx. We can simplify the (3+dx)^2 part into 9+6(dx) + dx^2 by multiplying it across. and the 3^2 into 9. Now what we have is:
9+6(dx) + dx^2 - 9 all divided by dx. The 9 and -9 cancel, leaving with us 6(dx) + dx^2 / dx ==> 6 + dx. Hope this helps.
This is the fundamental derivatives video. I wish he would highlight the videos by cornerstone concepts so that people could know what videos show the end-game of the topics.
123IGRACH That's how I saw it - on Khan. Even on the Website, the organization of the material isn't logically sequential (e.g. parts leading to a whole for mainstream taught concepts.)
Endgame? This comment was way ahead of its time 😮
dear lord im 19 and have been told that im smart and good at maths multiple times im even studying engineering and here i am trying to understand what a derivative is after calculating more than 202020030303 derivatives for the past 8 years
Thanks again Sal! These HD videos are great!
Mr Khan, thanks for reaching out to the young and hopeless. You'r awesome.
I wish you could do the whole course of Calculus 1 !! you are waaaaaaaaaaaaaaaaaaaaay better than my professor !!! :/
@Yvesiscool
It is, you're correct. Because he didn't take the general slope algebraically - he took it at x=3 - he got the derivative at that point (rather than the general derivative for all points which would be 2x). x=3 || 2x = 2(3) = 6
You just blew my mind
x = delta x... so think of it this way - (3+x)^2 is just (3+x) times (3+x). which is the same as 3(3+x) + x(3+x) which equals 9 + 3x +3x + x^2 - which gives you the 6x
ZuccheniMan thanks man you helped me out
THANK YOU!!! I know understand the calculus EC that my algebra 2 teacher gave us! There's no need of a tutor when ya got Khan Academy :))
Well because if he made it the limit as x --> 3 or 2, then that would be finding the slope of the SECANT line. So what he wants to do is make that distance between both points -- delta x -- minimal, so he make it zero. This give him the slope of the TANGENT line, which is basically the linear slope that is just exactly touching the point on the curve he wants to find the slope of. Hope this helps.
Respond to this video...
i love you man. your vids are percise, accurate and easy to understand.. i understood everything
thanks. ive seen several of your video series now and they are always helpful. you are a great teacher
ur voice is as calming as morgan freeman's
Thank you for these videos! Now I understand the section we are going over much more than before.
You are a Godsend! Thanks a million!
The day Khan Academy is in 4k is the day I will die!! :D
This was released on my birthday and I find it a bit coincidental that I'm watching this now
@zae4398 he's right, when you divide 6dx +(dx)^2 by dx, the dx divides both 6dx and (dx)^2, 6dx turns into 6 because dx/dx is 1,(some teachers say they cancel each other but not really) 1*6 is 6, while(dx)^2 turns into dx because (dx)^2 is dx*dx/dx which is dx. The numbers are NOT being multiplied. Like if I do 6+9 over 3, the 3 divides both 6 and 9, giving 2+3=6 and if I added first, 6+9=15, 15/3=5, so we get the same result. Know the difference between dividing #'x that are being + and #'s *.
lol its crazy how your name is sal, mine is too, everytime you say your name i get startled haha. thanks for the great videos, youve helped verify my choice of math as my major.
OMG i understand EVERYTHING NOW!!!! Thank you so much! I feel like a genius or something, haha.
thank you very much khan for taught me this way of thinking!
Good job sir 👍🏻
Wow this was incredibly useful!
brilliant. very well explained.
As i tell "present when needed most".Thanks
it all makes sense...sals da mvp
Awesome work !!
Thank you very much
OOOOOOH Now it all makes sense......
dat moment when the stuff your prof finally teaches is beyond the khan academy vids, Fuuuuuu!
thanks so much man , your efforts are surely great and for good sake too.
You're awesome, bro.
@Iislunatic We don't exactly divide by zero there. It's the limit of the slope when delta x approach 0.
Very good explanation.
Thank you!
dude i love your vids, theres a spot in heaven for you
@ViniciusFiocco Because when this: (6h+h^2)/h
Can be reduced to this: (6h/h) + (h^2/h)
The h's then cancel out and you get 6+h
one of the best videos superbbbbbbbbbbbbbbbb!!!!!!! gr8 video
really veryyyyyyyyyy helpful
Sir...
• How to find the slope of the tangent line to each curve when x has the indicated value.
• How to find the equation of the tangent line to each curve when x has a given value
What's even more amazing is that that is his handwriting on a computer! I mean, in real life on pencil and paper I'd say my handwriting is mediocre. But when I use paint or a drawing pad, my handwriting turns to crud. I wonder what his REAL handwriting looks like...:P
@Aitsu58 because the square of a number is it times itself so it would be (Dx X Dx)/Dx
thus giving you Dx
Simply Amazinggg
omg this is absolutely beautiful, i love math =')
Ur d best!!!
why does my calc book make this look like actual rocket science
•How to find the slope of the tangent to each curve when x has the indicated value.
• how to find the equation of the tangent line to each curve when x has a given value
@teakz89 He, is just calculating out the equation. (3 x deltax)^2 is
(3 x deltax)(3 x deltax) = 9 + 6 delta x + delta x ^2
you just saved my ass bro
its so funny how this video has half of part 1 views :D
+Peter Kvačkay I'm sure that results from the 'I suck at math' syndrome.
Ok... Im not learning calculus yet.. but it seems so confusing... lol slope slope slope slope.... im like :O
Oh, neow I understand
John smith *meow
The two people who dislike this video got an F on their math report card!
pretty easy
Sir I had one doubt..can we say that the curve had many different slopes..and so we chose a slope and made the equation of that slope as our tangent's equation..
great
How did you got the 6 delta x?
An not able to see ur writings
Ig at last it should be slope of :-
f(x)= 6 +∆X so keeping x as 3 answer should be 9 but 6 is written pretty sure it might be a simple mistake but wrote this comment to help anyone else or else understand my mistake
@7:45 Why can you “just set [delta x] to zero?” by using the limit? Isn’t this just hand waving as you convert some real distance Delta X - however tiny - to zero for the sake of convenience?
Wow....I feel like a nerd that I just laughed at that for about 10 seconds.
If the y axis is f(x), is the z axis f(x,y)?
this might be a dumb question, but if you set delta X to 0 doesn't that mean you had a 0 in the denominator earlier in the problem? isn't that meant to be off limits?
John Wilkes Booth if you're still looking for the answer watch the earlier limits videos
Assuming the following: Derivatives 2 > You
Search up the FOIL method
loooooove you.
yeah, in my understanding the "delta x" is just a variable in this equation.. isn't it? so you just have to copy it when adding, correct me if i'm wrong.
∆x is "variation of x" "∆" means "variation"
sir u explained very well and i learned now what is slope and derivative,
but please tell me in geometrical meaning of derivative why ∂x=dx but ∂y≠dy? plzz
In a way, but it's just a fancier way of saying "the slope of a point on a curve."
It seems this video cannot be viewed from within the Khanacademy site, ecxept in Bangla.
@iTreasons I failed calculus in high school. I did good in Trig and algebra. I went to college and studied physics. I ended up with a 3.8 GPA. I am trying to get a Masters in Physics with a concentration in quantum mechanics and particle physics. I am coming back to Calculus for personnel reasons. Unless you become a math major or engineer you don't need to completely understand calc. Like someone stated, just worry about GPA my friend, and breath... you'll be fine.
Nice bruh
My calculus is also much bad
And I'm trying harder to improve it at university level so I may gain a good gpa at the end of semester 😊
How is 3+dx squared 9+6dx+dx^2, I thought it was 9+dx^2, since you square the 3 and the dx?? Someone help please.
Why are we using the limit ?
We do it so that the gap h gets smaller and smaller and when h becomes 0 then the points x and x+h coincide, or in other words they become the same point and the line through that point becomes the tangent
does taking a derivative of function each time means we are taking its slope???
@DarklightALBANIA Δx is just a variable, if I give you a function of x
f(x) = 6 + x
then f(1) = 7
If I say f(Δx) = 6 + Δx
Then f(1) = 7, does that help?
I love sal
Maybe someone can help me out here. If we make delta x arbitrarily small, and we have to divide by delta x to find the limit, why does the limit not become arbitrarily large?
so derivative is the same as slope?
Where's the 6 delta x from at 4:22 please reply asap, have an exam in two days?
You do the FOIL method
what about the curves whose equations are not given.....i had a curve that said y=f(x) thats it and it was curve not a straight line
why 6dx +(dx)^2/dx isn't 6+(dx)^2 but 6+dx ?
how did it become 6 delta x? I thought you'll just have to multiply it to become 3 delta x. .. ?.?
by slope does he mean gradient
Sir, how many tangent lines does a point on a curve has?
It has only one tangent line i guess as it is one point only
@@maryamlina644 why not there s more than one?
@@jairoselin5119 cuz we are studying the slope of the curve at that particular point which means that we need only one tangent line 🤷♀️
@@maryamlina644 you're rgt.. but am asking how can you say that ua tangent is unique?