Exponential derivative visual

[@sathmika]

we should use digital interfaces to teach maths to students. anyway your channel is awesome....

[@dageustice]

f(x)=0 also has itself as its derivative, but no one seems to care at all ☹️

[@MathVisualProofs]

Is true :)

[@catmacopter8545]

f(x)=0 is like objectively the least interesting function

[@THEMathHacker-121]

@@catmacopter8545 really? Quite useful in solving quadratics, cubics…

[@dageustice]

@@catmacopter8545 Are you sure about that? Integrate it a few times and you can produce ANY polynomial. Give me a different function that can do the same 😤😤😤

[@XenophonSoulis]

All functions of the form f(x)=ce^x have themselves as their derivative and this includes the case for c=0.

[@dospiir8045]

her : i can change him him :

[@stevenlaczko8688]

actually genius

[@justinliu7788]

her: ln

[@ishaanroy2436]

😂

[@lillii9119]

By the definition of derivative, we find that (a^x)' = (a^x)(a^h - 1)/h when h -> 0, Let e be the number so that (e^h - 1)/h = 1, e^h - 1 = h e^h = 1 + h e = (1+h)^(1/h) By definition, this is the only real number that, as the base of an exponential function, gives that function itself as a derivative

[@smikenickleby9839]

this is a nicer way to look at it, because the earliest "definition" of e is the series that generates from (1+1/n)^n as n -> infinity (n = 1/h). What the standard calculus approach tells us (that you wrote down), is that e is the only number that ensures a^h - 1 is approaching with the same magnitude as h, when h is approaching zero. It has more clarity from discrete math perspective, when h is small but non-zero: a^h - 1 is *the closest* to h, when a = e.

[@gilly4487]

I hate my Calc 1 class rn but this video genuinely helped me. Thank you.

[@kitspapp]

i remember my maths professor showed us this proof in intro calc, my mind was blown

[@lyrimetacurl0]

I check my brain works ok by working out y'=y. I get y= plus or minus e^x

[@speakingsarcasm9014]

​@@lyrimetacurl0 you get y=ce^x where c is a constant...

[@pagoluharshavardhangowdme1423]

It also has another interesting property. Although it's slope increases exponentially it's tangents intercept on y axis increases linearly with x i.e the intercept of tangent at x=1 is 0, at x=2 is 1, at x=3 is 2.... And so on

[@aryaganne9364]

basicaally the way log function works (in this case natural log, but for logs you can translate multiplacation of powers into sums ex log base 2 of (2^3)*(2^2) = 3+2)

[@skyzm212]

Another intereating method is using taylor's expansion series to prove that for all integer n since e^x=1+x+x^2/2+...x^n/n .then it's derivative would remain the same neglecting the rest since we're doing a derivative

[@bijipeter1471]

Thank you,sir

[@Umlaut95]

Thank you! This helps with comprehending calculus

[@omerutkuerzengin3061]

So nice explanation.

[@loohooi6545]

In addition,the difinite integaral(the area below the curve) from negative infinity to any x value of that point is also exactly to its y-coordinates.

[@virushk]

NICE

[@christianmosquera9044]

excellent video

[@MathVisualProofs]

Glad you liked it!

[@elnetini]

The next step would be to show the area under the curve, which is the same value as the slope at any given point

[@pelayomedina2174]

That's literally how e^x is defined

[@samueljehanno]

😂

[@isavenewspapers8890]

It's just one possible definition.

[@king_halcyon]

​@@isavenewspapers8890main definition nonetheless

[@jesusnoagervasini8207]

f(x)=0: Look at what they have to do to mimic a fraction of my power

[@Cow.cool.]

f(x)=0 isnt an eigenfunction of the derivative operator as eigenfunctions must not be zero. Also f(x)=0 is just an extreme case of the exponential function of the form ae^bx

[@logosking2848]

Not by coincidence, but by definition

[@samueljehanno]

😂

[@AL73250]

If we define that 3^X equals its derivative, that wont make it true. Its only true for 2.71828... That cannot be defined, it is just given, from our perspective that's a coincidence

[@omkarnagarhalli5217]

@@AL73250we define that interesting number, 2.71828, as e, because of this. Since f(x) = 2.71828^x is the only function that is the same as its derivative (along with the trivial x = 0), that number holds a special place in math

[@AL73250]

@@omkarnagarhalli5217 Sounds more like just calling it e. Defining should involve a little more than giving a name right?

[@omkarnagarhalli5217]

@@AL73250 yes and the exact definition of e necessitates that its derivative is itself

[@Fire_Axus]

fun functions

[@Khalid.115]

Fun(1+ctions)

[@shadowshibe5962]

Oh my God I so needed this

[@thefirsttrillionaire2925]

Wow I’ll never forget this now 😮

[@RunningOnEmpty-f7r]

It is quite easy to understand once it's visualused like this, but personally I really like the other explanation of its consistency which involves Taylor series. You see, e^x can be expressed as an infinite sum of x^n/n! so it goes like this: 1+x + x²/2!... and so on.... So if you find a derivative of this you'll get the exact same thing, and I really *love it*

[@korakatk318]

How did we find e in the first place? Pi and e are so famous but school never taught me the history behind e

[@MathVisualProofs]

This video is one way of coming up with e. You look for a function that is it’s own derivative. All Exponentials almost have this feature but only one is exactly equal to its derivative (others are multiples of their derivative)

[@py8554]

For me the most intuitive definition for e is the limit of (1 + 1/n) ^ n as n approaches infinity. The equation arises when we consider the situation of increasing the compound interest evaluation frequency while keeping the interest rate fixed.

[@korakatk318]

@@MathVisualProofs interesting, but thanks for replying!

[@korakatk318]

@@py8554 thanks!

[@Ceereeal]

It is also the infinite sum of the reciprocals of all the factorials: 1/0! + 1/1! + 1/2! + 1/3! + 1/4! + …

[@grimanium]

not only that, that also means the area under the curve is equal to the curvature which is equal to y :)

[@nikolatesla702]

Sir please make a video on LEFT HAND DERIVATIVE and RIGHT HAND DERIVATIVE. in the same way.

[@Tiggster-qr8mw]

Multiply e^x by any constant and the same thing still applies

[@buzeqeshjeetrishtuar]

Yeah no shit Sherlock that's how derivatives work

[@georgecop9538]

we had to find out the derivatives of e^x, sin and cos using fundamental limit and the definition of the derivative. This is more easier, though it is less rigorous.

[@MathVisualProofs]

yes. this just gives the idea that those are correct. you still need to work out the limit definition details :)

[@ondriktv4007]

try to make visual calculating roots for quadratic formula :) that would be amazing!

[@MathVisualProofs]

It's essentially this: kzbin.info/www/bejne/imGrqop_e8emfrM or this: kzbin.info/www/bejne/mHbXlX6dl9SgidE

[@FatalKill]

This is why exponential functions are peak.

[@badabing3391]

show the values in terms of e

[@dawne2780]

Woah this is so much easier to remember now haha

[@lucahermann3040]

Don't take it one step further. Go one step backwards. The tangent always intersects the x-axis at x-1.

[@littleantukins4415]

e^x is the goat of all functions

[@wiggles7976]

This video is sort of circular. Your givens included all the points of the graph f(x) = exp(x) which you used. How did you obtain them in the first place?

[@MathVisualProofs]

not really circular. This isn't proving anything. It is just a short showing the derivative graphing out. You can in fact show that the derivative of exp(x) is itself, but that's for another video :)

[@tied515]

The word "proof" doesn't occur in the video title... but the channel is called "visual proofs", which is misleading. Imo this is not even visual, because we cannot "see" that the slope equals the ordinate, we just read the output of the computer. And the computer doesn't produce this output by a slope computation.

[@DhesCatolos]

By 1-1 f(x) y² x⁴= derive 3

[@mananagrawal6855]

I mean, that's how e is defined.

[@marciorjusto]

What the difference between Power ("Potentiation") and Exponentiation? Why they have different inverse operations?

[@Krishna_00123]

Sir light ka speed vaccum me kyun constant hai??❤❤

[@cdkw8254]

e^x the goat

[@sailonwatt7557]

Log: I can change him.

[@7Cetus7]

I was expecting to se the exponential derivative of a function in this short...

[@androkguz]

The intersection of the people that know what a slope/derivative is but don't know that the derivative of e^x is e^x just be very very small.

[@karlmartell7600]

This is always extremely funny to me, because the literal Definition of e^x ist that it is a function who's Derivative ist itself and which has the value of 1 at x=0. Which is mostly there so the solution is not f(x)=0. So EVERY proof for that is literally just a proof that nobody Made an error in devising that function.

[@jaydenchan3568]

Who has not be amazed to learn that the function y = ex, like a phoenix rising again from its own ashes, is its own derivative? David Darling

[@samueljehanno]

e^x not ex

[@madamada219]

What I think is weird is that f(x)=sin(x). Is same as f''(x)

[@jan-lukas]

​@@madamada219nah f(x)=sin(x)=f''''(x) f''(x)=-sin(x), right?

[@GoldenAdrien]

Is there any other functions that do this? It is something that has been a curiosity of mine since I first learned derivatives

[@juliavixen176]

I believe that this is the only non-trivial function on the real number domain. (The trivial functions being zero or scaled copies of eˣ)

[@carultch]

It's the only function that does this. You can prove it with the simplest differential equation you can do. dy/dx = y dy/y = dx integral dy/y = integral dx ln(y) = x + c y = e^(x + c) y = e^c*e^x Let capital C = e^c, so it becomes: y = C*e^x The only other functions that do this are constant multiples of e^x.

[@rawat789]

Me understanding this at 34 🤯🤯

[@davidmurphy563]

I've been trying to get the function s-e^x for years, asked my wife but she says she's got a headache.

[@zionsky3342]

I'm tryna plot that function but I just get a straight line

[@zionsky3342]

Oh wait no it's okay I figured it out :)

[@Nightlife_Offical]

so when x = -infinity, then y=0, (i know it should be undefined but still...) then 0=e^-infinity

[@hiredfiredtired]

yea

[@Ducksaregreat]

Ah yes, I know what all of this means.

[@Franzzz_E]

Hey man i just got a mental breakdown because i saw that apparently 0.999... = 1 could you make a video on that?

[@MathVisualProofs]

It’s in the queue!

[@Franzzz_E]

Tnx

[@KrisPBacon69]

Notice that 0.999... is the same thing as lim(x->inf) 1 - 1/(10^x). This is because if you have x=1 it's 1 - 1/10 = 9/10 = 0.9, if you have x=2 it's 1- 1/100 = 99/100 = 0.99, and as x approaches infinity it will become 0.999... Since 1/(10^x) approaches 0 as x approaches infinity, the value of the expression is just 1 - 0 = 1, hence 0.999... = 1.

[@wiggles7976]

Why wouldn't it be the case that 0.999... = 1? What is their difference? 0.000...0001? If that is the difference, then try to express the number in standard notation. It's equal to 1*10^(-c) for some c. What is c? Does c belong to the set of real numbers? No, it does not.

[@assimsendo]

Show show....

[@Robin-Dabank696]

I feel like this would be true for anything raised to the power of x...

[@MathVisualProofs]

It’s close. But you get am extra scale factor of ln(b) for b^x

[@skc4188]

A surprisingly long video for just a "short". =P

[@SPIDERMAN-q6r7g]

I didn't understand a word. I'm just here cause math's 😎

[@CombineWatermelon]

The ice fairy lookin hella confused after this one

[@alphanahmetunsal]

Se^x=e^x😮

[@holthuizenoemoet591]

i love you

[@nothingtoseehere2189]

How is this visual it’s just saying look it’s true now let’s graph it

[@3._1415]

I'm sorry but you lost me after two seconds... but it looks cool lol

[@tcadityaa]

This is more of an observation...this explains nothing...

[@MathVisualProofs]

It’s meant to show how the derivative is obtained from a function by visual example. In the queue is visual def of how we define derivative in general.

[@sirknightartorias68]

No I don't see that... Its just you've said things that are already there.. no visualisation whatsoever..

[@nathanderhake839]

This isn’t a proof

[@mcpecommander5327]

Doesn’t seem like a proof

[@gioxmemes8285]

Graft it

[@Dante-420]

I'm not convinced by this proof. It doesn't show why the slope of the tangent at (x,y) is equal to y. It just says so in a circular logic kind if way. (And yes, I have seen more convincing/actually rigorous proofs, and I understand that this is one of the definitions of e, but this visual "proof" doesn't demonstrate any of that)

[@MathVisualProofs]

Yeah. This isn’t proof. Just visualization as the title says.

[@jadencoles380]

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[@torb1trick415]

this “derivative” thing is dumb. why are you plotting the slope on a graph? what’s the point? some guy probably thought he was smart when he thought of that. i hate math and when smartassery like this shows up i can’t stand it.

[@cheekywombat9208]

You've never driven a car, have you? 😂

[@someguy1428]

This is literally used in everything, even the KZbin algorithm that push this shorts to you

[@torb1trick415]

@@someguy1428 as of you couldn’t do that another way more easily, it’s over complicating things for nothing

[@hiredfiredtired]

@@torb1trick415 Try. Try thinking of another way do to this.

[@insouciantFox]

I thought you were going to graph the literal exponential derivative, as in e^Df(x)= f+f'+½f"+⅙f"'+...