Proof: Derivative of e^x is e^x

In this video, we follow the definitions of the derivative and the number e to prove that the derivative of e^x is indeed equal to e^x.

#Calculus #Differentiation #Exponential

Thanks for watching. Please give me a "thumbs up" if you have found this video helpful.

Please ask me a maths question by commenting below and I will try to help you in future videos.

[LarsSandgren]

Me during the first part of the video: Isn't he just going in circles?.. Me during second part of the video: We are just going in circles.... Me during final part of the video: WE ARE SUPPOSED TO GO IN A CIRCLE!!! I don't know who made this proof but, hot diggity it is jaw dropping

[bejjwkwnsbq]

Thank you very much. This is the best proof I've seen so far. The other ones here on YouTube usually involve some kind of unsatisfactory implicit differentiation. Not this one!

[uvuvwevwevweonyetenyevweug]

How about we say that Since e^x is optimal growth.
And if we derive this we find
0+1+x/1!+x^2/2!+x^3/3!... And so on. Hence e^x is equal to its derivative.

[spudhead]

Right at the start you have circular reasoning. You see, d/dx e^x = e^x IS a definition of e. Such that e is the only value of n that satisfies d/dx n^x = n^x. All definitions are equivalent, so what you are doing by taking the definition of e at the start is effectively saying: to prove d/dx e^x = e^x, we first start by asserting d/dx e^x = e^x. You see the problem? Using the limit definition of e is exactly the same as using the derivative definition of e. But more than that, YOU CANNOT PROVE A DEFINITION, if you could you wouldn't need it to be a definition in the first place. No matter what you do, you simply cannot prove it without some kind of circular reasoning. Maybe we will develop the maths tools someday that allow us to prove the value of e, but until then, we can only define it.

[Kzaman-ygbr]

6:50 can I apply L Hospital rules because it is now 0/0 form.

[geraldvaughn]

How did you bring the limit inside the natural log? What rule is that?

[Xayuap]

this one was satisfactory
I rather predefine e than predefine a magical exp

[NumbToons]

What a beautiful video and presentation

[timding]

At 3:56, I do not understand why the e to the power of x is "independent" of the limit and can go to the front of the whole limit/right hand side. Many videos say the same thing but never explain that. Is there a video that explains that? Thank you!

[ksmyth]

Thanks for the video. I think it was well presented. But I wouldn't say it is based on first principles, because you have already made an assumption about e. Is it not possible to only assume there is a function such that f''(x) = f(x) and then derive its characteristics? The function would be k^x. One would then have to determine a value for k. Or is this not possible? I suppose as with most theorems or derivations, the problem is deciding where the buck stops.

[philw]

Thank you, I found this really clear. Including approaches that didn’t work was very helpful too (I’ve watched other ‘proofs’ that evaluated 0/0 as 1 with no justification).

[oscarludwinmalaveranarcaya]

Best proof of this derivative, others imply knowing what Taylor series is but that's ridiculous if we're working with the definition of derivatives

[ramoimas]

You are the man. You are him. You know where the dog is, you know victoria's secret. You can see john cena. My pronouns are he not him. Cuase I can never be you.

[saravanarajeswaran]

on the other side, any function say a^x, the derivative would become a^x times limit as h approaches 0 of a^h - 1/h, which gives some random irrational number (which ln a), so we would ask, so is there any number, so that we take derivative the limit becomes 0, yes that is e, so most likely it is one of e definition you can say from another persepective

[doyourealise]

amazing! thanks a ton, subscribed! wil be finishing all of your videos.

[renesperb]

If the chain rule and the derivative of the log-function is known, then there is a very fast and simple proof: set y(x) = e^x and
use that d/dx(ln y(x)) =1/y(x) *y'(x), but by definition ln(e^x) = x, hence we have 1/y(x) * y'(x) = 1, that is (e^x) ' = e^x.

[timm]

the exponential function is defined as that unique function that is its own derivative. proof is not necessary.

[nynthes]

Hi, what program do you use to make these videos?

[MundiaMukelabai]

the video is well explain but I only have one question. can you show us where and why e is defined as given in the video because we just adopted it and not shown it also how e is defined by the supposed given limit

[samuelprieto]

Coulda skipped a bunch of steps but still excellent proof ;)