Brilliant!! I just looked ahead to learn a little more about this function and as soon as I saw the absence of a textbook proof of the derivative of e^x I went straight to YT. Thanks Khan Academy!
Dear Khan, I love this video so much. Your passion for teaching just shines in how amazingly you explained this beautiful phenomenon of the number e. God bless you for being a great teacher!!! =)
you could think of it as if he plugged the limit to both the numerator and the denominator. since the limit does not affect 1(since its an integer), it only affected the denominator.
Strictly speaking, lim(x->a)(f(g(x))) = f(lim(x->a)(g(x))) if f(x) is continuous on a given interval. Its kinda the definition of a continuous function, but i believe there is a way to show that 1/lnx is continuous on (0;1)
What an elegant proof! I'd like to ask just one thing: from 1:07 to 1:19, wouldn't the slope of the tangent line be equal to "exp x" instead of just "x"? I also might have misheard something, so any clarifications would be appreciated :)
What about Exp(x) = Limit on n of S(x,n), where S(x,n) = Summation from 0 to n of x**k/k!. Because S(x,n) converges uniformly on any finite interval, we can do the derivative before the limit. And because d(S(x,n)/dx = S(n-1,x), the limit of the derivative is the same than the original limit. Lim S(n,x) can be the definition of e**x
Well by now you probably now but anyhow, e=limu->oo (1+1/u)^u N=1/u since u approaches N is 0 and 1/N will be infinity(positive infinity since N can only approach 0 from the right since ln only takes positive input)
@practicalaccount2069 yeh thank you, been through cal 1 for a while now, seeing this comment gives me some nostalgia, now I don't remember much about this subject anymore, but I did remember figuring this out not for too long after commenting this, sadly no one replied
At 2:52 , I do not understand why the e to the power of x is moved to the front of the whole limit. Many videos say the same thing but never explain that. Is there a video that explains that? Thank you!
Thanks for not skipping steps Sal! : )
"Who has not been amazed to learn that the function y = ex, like a phoenix rising from its own ashes, is its own derivative?" - Francois le Lionnais
e^x **
Finally someone who doesn't skip the resolution of the limit or use d(lnx)/dx to prove it
Okay... that was the best thing I’ve seen all night! Very well done!
Brilliant!! I just looked ahead to learn a little more about this function and as soon as I saw the absence of a textbook proof of the derivative of e^x I went straight to YT. Thanks Khan Academy!
Your explanation is really good but your English is great ! For me , that makes all the difference .
Great proof its so simple and yet rigorous enough :D
Dear Khan, I love this video so much. Your passion for teaching just shines in how amazingly you explained this beautiful phenomenon of the number e. God bless you for being a great teacher!!! =)
at 7:15 you change the ln((1+n)^(1/n)) to the ln of the limit of ((1+n)^(1/n)). What property is that?
it's just a basic limit property.
you could think of it as if he plugged the limit to both the numerator and the denominator. since the limit does not affect 1(since its an integer), it only affected the denominator.
Strictly speaking, lim(x->a)(f(g(x))) = f(lim(x->a)(g(x))) if f(x) is continuous on a given interval. Its kinda the definition of a continuous function, but i believe there is a way to show that 1/lnx is continuous on (0;1)
@@dolevgo8535 When you do lim(x->a)(1/x), you end up with lim(x->a)(x^-1). Why in this proof you could take the limit of the denominator only?
Thankyou so much sir! It all makes sense now..
this is beautiful ❤
What an elegant proof! I'd like to ask just one thing: from 1:07 to 1:19, wouldn't the slope of the tangent line be equal to "exp x" instead of just "x"?
I also might have misheard something, so any clarifications would be appreciated :)
Yes, it would
It would be equal to y which is exp(x)
Thank you so much sir , awesome video finally I was able learnt it alhamdulillah
Yup I got that tingly feeling
What about Exp(x) = Limit on n of S(x,n), where S(x,n) = Summation from 0 to n of x**k/k!. Because S(x,n) converges uniformly on any finite interval, we can do the derivative before the limit. And because d(S(x,n)/dx = S(n-1,x), the limit of the derivative is the same than the original limit.
Lim S(n,x) can be the definition of e**x
You teach soo well!!!!!!
Kahn you're a true hero
HitchHiker31 its KHAN not KAHN.
wait? at 0:23 why e=lim(1+n)^1/n when n approaches 0??? i haven't seen that definition in anywhere?
Well by now you probably now but anyhow, e=limu->oo (1+1/u)^u
N=1/u since u approaches N is 0 and 1/N will be infinity(positive infinity since N can only approach 0 from the right since ln only takes positive input)
@practicalaccount2069 yeh thank you, been through cal 1 for a while now, seeing this comment gives me some nostalgia, now I don't remember much about this subject anymore, but I did remember figuring this out not for too long after commenting this, sadly no one replied
great improvements in these introductory differential calculus videos :)
keep it up!
At 2:52 , I do not understand why the e to the power of x is moved to the front of the whole limit. Many videos say the same thing but never explain that. Is there a video that explains that? Thank you!
It's unaffected by the process of obtaining the limit, so it's able to be treated as a constant factor :)
Genius!!
WHY DIDN'T YOU POST THIS 3 DAYS AGO?!?!
Still great video and thanks
why is 'e' defined in such a way? where does this definition came from?
Please solve e^x^2 with first principle
Please no more crosshairs.
ôi mẹ ơi !! Hay quá :))) Số e thật tuyệt vời, thank you so much !!!!
so fancy.... :,)
noice
😂😮😅😊you cant do what I can unless I teach you and I wont😢 that you😅thats me Bye🎉your heads
First
Downvote! e is a LETTER not a number. This guy doesn't know what he's talking about!
e is a number.
its euler number app 2.71....
lol what did i just read, my eyes got cancer
Greatest joke of the century!
you don't know exponential funtion
Title of this video is: Proof: The derivative of 𝑒ˣ is 𝑒ˣ