/sci/ - Science & Math » Thread #10759121

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Anonymous Wed 26 Jun 22:25:04 2019 No.10759121 View Reply Original Report

Quoted By: >>10759135 >>10759148 >>10759189 >>10759285 >>10759408 >>10759648 >>10759674 >>10759702 >>10759789 >>10759804 >>10761810

If you cannot prove this, you are a brainlet and need to leave this board immediately.

Anonymous

Anonymous Wed 26 Jun 2019 22:28:34 No.10759129 Report

Quoted By: >>10760175

Waste of time. Study theology instead

Anonymous

Anonymous Wed 26 Jun 2019 22:32:27 No.10759135 Report

Quoted By: >>10759164 >>10759252 >>10759454 >>10759647 >>10761077

>>10759121
First result is true by definition
Second result is proved using the fucking chain rule
You arent smart for passing calc 1

Anonymous

Anonymous Wed 26 Jun 2019 22:34:53 No.10759143 Report

Quoted By:

trivial

Anonymous

Anonymous Wed 26 Jun 2019 22:36:16 No.10759148 Report

Quoted By:

>>10759121
have sex

Anonymous

Anonymous Wed 26 Jun 2019 22:40:30 No.10759164 Report

Quoted By: >>10759181 >>10759226

>>10759135
He means prove the definition

Anonymous

Anonymous Wed 26 Jun 2019 22:44:35 No.10759181 Report

Quoted By:

>>10759164
>prove the definition
?

Anonymous

Anonymous Wed 26 Jun 2019 22:46:24 No.10759189 Report

Quoted By: >>10759201 >>10759427

>>10759121
Not always true, if e=1 then the derivative is 0

Anonymous

Anonymous Wed 26 Jun 2019 22:49:24 No.10759201 Report

Quoted By:

>>10759189 damn you've cracked the case

Anonymous

Anonymous Wed 26 Jun 2019 22:58:54 No.10759226 Report

Quoted By: >>10759232 >>10759278 >>10759411 >>10759478 >>10759611 >>10760120 >>10761716

>>10759164
>He means prove the definition
I'm not a "he".

Anonymous

Anonymous Wed 26 Jun 2019 23:01:00 No.10759232 Report

Quoted By: >>10759234

>>10759226
I forbid gender to be brought up here, we shall refer to each other as 'the right honourable person'

Anonymous

Anonymous Wed 26 Jun 2019 23:01:34 No.10759234 Report

Quoted By:

>>10759232
sure is summer here

Anonymous

Anonymous Wed 26 Jun 2019 23:10:11 No.10759252 Report

Quoted By:

>>10759135
It can be proven either by taylor series or by converting it to ln format.

It can be found 5 min of googling. Why use brainpower of reinventing the wheel?

Anonymous

Anonymous Wed 26 Jun 2019 23:22:51 No.10759278 Report

Quoted By:

>>10759226
>she

Anonymous

Anonymous Wed 26 Jun 2019 23:27:03 No.10759285 Report

Quoted By:

>>10759121
>Posting trivial stuff
>look at me i learned some high school calculus i am good at math
GTFO fucking moron

Anonymous

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Anonymous Thu 27 Jun 2019 00:15:09 No.10759404 Report

Quoted By: >>10759424 >>10759441

>no one has proven it yet
>most posters only proved they have no idea what the fuck they're talking about
Everyone itt needs to gtfo my board

Anonymous

Anonymous Thu 27 Jun 2019 00:19:33 No.10759408 Report

Quoted By:

>>10759121
Watch 3blue1brown’s series on calculus on calculus and try to understand everything he says. Get a notebook and make sure you realize fully where all the rules are drawn from.

Anonymous

Anonymous Thu 27 Jun 2019 00:21:59 No.10759411 Report

Quoted By:

>>10759226
By using a post to make that statement, you proved why people hate when a “she” tries to discuss anything. If you’re honorable, you’d have kept quiet. Also, I may have fell for a bait.

Anonymous

Anonymous Thu 27 Jun 2019 00:26:33 No.10759422 Report

Quoted By: >>10759424 >>10759454 >>10759647

>1
true by definition of the exponential function
>2
f(x)=a^x=exp(ln(a^x))=exp(x*ln(a))
=> f'(x)=ln(a)*exp(x*ln(a))=ln(a)*exp(ln(a^x))
=> f'(x)=ln(a)*a^x

Now stop being underage

Anonymous

Anonymous Thu 27 Jun 2019 00:27:36 No.10759424 Report

Quoted By:

>>10759404
See
>>10759422

Anonymous

Anonymous Thu 27 Jun 2019 00:30:05 No.10759427 Report

Quoted By:

>>10759189
Based

Anonymous

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Anonymous Thu 27 Jun 2019 00:39:16 No.10759441 Report

Quoted By: >>10759456 >>10760886

>>10759404

Anonymous

Anonymous Thu 27 Jun 2019 00:41:25 No.10759454 Report

Quoted By: >>10759479

>>10759422
>true by definition of the exponential function
Why did you feel the need to prove you're a mouth-breathing retard a second time >>10759135

Anonymous

Anonymous Thu 27 Jun 2019 00:41:45 No.10759456 Report

Quoted By:

>>10759441
sorry for making you break your necks to read it

Anonymous

Anonymous Thu 27 Jun 2019 00:49:33 No.10759478 Report

Quoted By: >>10759484

>>10759226
You’ll never be a woman.

Anonymous

Anonymous Thu 27 Jun 2019 00:51:21 No.10759479 Report

Quoted By: >>10759492 >>10759668

>>10759454
But anon, "the function whose derivative is equal to itself" IS the definition of the exponential function. The Taylor expansion follows from this, not the other way around.

Anonymous

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Anonymous Thu 27 Jun 2019 00:53:26 No.10759484 Report

Quoted By:

>>10759478
>transplaining

Anonymous

Anonymous Thu 27 Jun 2019 00:56:30 No.10759492 Report

Quoted By: >>10759512

>>10759479
They are equivalent once you supply a boundary condition) but the Taylor series defines the exponential over the entire complex plane and over many other rings.

Anonymous

Anonymous Thu 27 Jun 2019 01:04:24 No.10759512 Report

Quoted By:

>>10759492
>but the Taylor series defines the exponential over the entire complex plane and over many other rings
based

Anonymous

Anonymous Thu 27 Jun 2019 01:16:57 No.10759537 Report

Quoted By: >>10759546

Why does Euler's identity?

Anonymous

Anonymous Thu 27 Jun 2019 01:20:01 No.10759546 Report

Quoted By:

>>10759537
Uses the definition that the derivative of exp()= exp()

Sage

Sage Thu 27 Jun 2019 01:22:25 No.10759550 Report

Quoted By:

God I hate underage retards

Anonymous

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Anonymous Thu 27 Jun 2019 02:01:48 No.10759611 Report

Quoted By:

>>10759226
...sure

Anonymous

Anonymous Thu 27 Jun 2019 02:24:06 No.10759647 Report

Quoted By:

>>10759422
>>10759135

2 is an incomplete proof. It remains to prove that the rules for manipulating expressions of exponents and logarithms remain the same when they involve irrational numbers. Turn to baby Rudin, Ch1, exercises 6 and 7.

Anonymous

Anonymous Thu 27 Jun 2019 02:25:35 No.10759648 Report

Quoted By: >>10759651

>>10759121
ln(e^x)=1

Anonymous

Anonymous Thu 27 Jun 2019 02:26:46 No.10759651 Report

Quoted By:

>>10759648
ln(e^x)=x

Anonymous

Anonymous Thu 27 Jun 2019 02:30:08 No.10759659 Report

Quoted By:

4/8(16)=72/125

Anonymous

Anonymous Thu 27 Jun 2019 02:37:45 No.10759668 Report

Quoted By:

>>10759479
>But anon, "the function whose derivative is equal to itself" IS the definition of the exponential function.
you need to include and f(0) = 1
otherwise the 0 function also counts

Anonymous

Anonymous Thu 27 Jun 2019 02:46:17 No.10759674 Report

Quoted By:

>>10759121
good thread demonstrating why you should try to prove the OP before you respond to these kinds of threads

Anonymous

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Anonymous Thu 27 Jun 2019 03:16:23 No.10759702 Report

Quoted By: >>10759725 >>10759781 >>10759843

>>10759121
$\frac{dy}{dx} = y$
$\int{\frac{1}{y}dy} = \int{1dx}$
$ln(y) = x$
$y = e^x$

$y = a^x$
$ln(y) = x \cdot ln(a)$
$\frac{dy}{dx}ln(y) = \frac{dy}{dx}x \cdot ln(a)$
$\frac{1}{y} \cdot \frac{dy}{dx} = ln(a)$
$\frac{dy}{dx} = y \cdot ln(a)$
$\frac{dy}{dx} = a^x \cdot ln(a)$

GG EZ

Anonymous

Anonymous Thu 27 Jun 2019 03:31:57 No.10759725 Report

Quoted By:

>>10759702
https://www.albert.io/blog/derivative-of-lnx-natural-log-proof-review/
As a supplement to this.

Anonymous

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Anonymous Thu 27 Jun 2019 03:50:29 No.10759760 Report

Quoted By:

/sci/ BTFO by Tibees

https://youtu.be/_2-3uY0IAro

Anonymous

Anonymous Thu 27 Jun 2019 04:04:45 No.10759781 Report

Quoted By: >>10759790

>>10759702
The first proof is invalid. The general solution of y'=y is a*e^x, not e^x.

Anonymous

Anonymous Thu 27 Jun 2019 04:10:03 No.10759787 Report

Quoted By:

The Taylor series can prove this pretty quickly

I'll leave this exercise up to the reader

Anonymous

Anonymous Thu 27 Jun 2019 04:12:57 No.10759789 Report

Quoted By:

>>10759121
Proving stuff like this is the reason I left math related field, my logic and mathematical reasoning was trash. I know 2+2 = 4 and can understand why the derivative for X^2 = 2x but that's as far as I went with math. Applied math was 2 hard 4 me.

Anonymous

Anonymous Thu 27 Jun 2019 04:13:35 No.10759790 Report

Quoted By: >>10759802

>>10759781
Fine, I forgot to add the +C after integrating.
either way, the proof for the first follows trivially from the proof for the second.
$\frac{d}{dx}a^x = ln(a) \cdot a^x$
$\frac{d}{dx}e^x = ln(e) \cdot e^x$
$\frac{d}{dx}e^x = e^x$

Anonymous

Anonymous Thu 27 Jun 2019 04:24:57 No.10759802 Report

Quoted By: >>10759808

>>10759790
You would do better proving the exponential function is the derivative of the exponential function without relying on its inverse or integration

Anonymous

Anonymous Thu 27 Jun 2019 04:25:35 No.10759804 Report

Quoted By:

>>10759121
Consider the initial value problem $y' = y ,\ y(0) = 1$ . By the general theory of differential equations, the solution exists, is unique and its domain of definition is the entire real line. Let us denote this solution as $y = f(x)$ . Note that $f(x)$ is always non-0 as the differential equation is linear and as $f(0) = 1 \neq 0$ .

Now let $a$ be an arbitrary real number and let $g(x) = \frac{f(x + a)}{f(x)f(a)}$ . So $g'(x) = \frac{f(x + a)}{f(x)f(a)} - \frac{f(x + a)}{f(x)f(a)} = 0$ which implies $g \equiv g(0) = 1$ . Equivalently, $f(x + a) = f(x)f(a)$ .

Define $e$ to be $f(1)$ . It is then easy to see (using the property derived above) that for any rational number $m/n$ , $f(m/n) = e^{m/n}$ and hence by continuity, $f(x) = e^x$ .

Anonymous

Anonymous Thu 27 Jun 2019 04:28:18 No.10759808 Report

Quoted By: >>10759859

>>10759802
Why bother doing it like that when my way is easier?
Yes, I could do it from the definition of derivative and the taylor series expansion and yadda yadda yadda but I don't see a reason to.

Anonymous

Anonymous Thu 27 Jun 2019 05:10:12 No.10759843 Report

Quoted By:

>>10759702
yeah but prove integral 1/y dy = ln(y) + C

Anonymous

Anonymous Thu 27 Jun 2019 05:23:01 No.10759859 Report

Quoted By:

>>10759808
you can do it without using either, and its the most instructive proof by far.

Anonymous

Anonymous Thu 27 Jun 2019 07:01:47 No.10760043 Report

Quoted By:

This is a (very succesful) bait thread but y'= y is just one definition of the exponential. For a more "advanced" use, you will want to use its series definition.
Not that it matters as the result is equally as trivial.

Anonymous

Anonymous Thu 27 Jun 2019 08:32:09 No.10760120 Report

Quoted By: >>10760178

>>10759226
kys faggot

Anonymous

Anonymous Thu 27 Jun 2019 09:02:34 No.10760175 Report

Quoted By:

>>10759129
kek

Anonymous

Anonymous Thu 27 Jun 2019 09:04:08 No.10760178 Report

Quoted By: >>10760869

>>10760120
>faggot
Why the homophobia?

Anonymous

Anonymous Thu 27 Jun 2019 16:19:52 No.10760869 Report

Quoted By:

>>10760178
Confirmed faggot

Anonymous

Anonymous Thu 27 Jun 2019 16:25:43 No.10760886 Report

Quoted By:

>>10759441
>Using a calculator
Gtfo

Anonymous

Anonymous Thu 27 Jun 2019 18:20:28 No.10761077 Report

Quoted By: >>10761782

>>10759135
>First result is true by definition
No it isn't. We can define a function exp(x) by D exp(x) = exp(x), you're asked to prove exp(x) = e^x, where e = 2.718...
After all, exp could be any function, it doesn't have to be an exponential, and the base doesn't have to be e: you have to prove that it is.

Anonymous

Anonymous Thu 27 Jun 2019 21:47:00 No.10761716 Report

Quoted By:

>>10759226
tits or gtfo

Anonymous

Anonymous Thu 27 Jun 2019 22:17:06 No.10761782 Report

Quoted By: >>10761799

>>10761077
Yes it is. The taylor expansion of f(x)=exp(x) follows straight from the definition that f'(x)=f(x). Then plug in 1 to get f(1). Guess what number we get? 2.718... which we name e.

Anonymous

Anonymous Thu 27 Jun 2019 22:23:07 No.10761799 Report

Quoted By: >>10761851

>>10761782
It really doesn't, the solution to y'=y is ae^x

Anonymous

Anonymous Thu 27 Jun 2019 22:27:19 No.10761810 Report

Quoted By:

>>10759121
You set the bar pretty low for brainless status.

Anonymous

Anonymous Thu 27 Jun 2019 22:50:13 No.10761851 Report

Quoted By:

>>10761799
Forgot about that.
"The function f such that f(x)=f'(x) and f(0)=1" uniquely defines f(x)=exp(x

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