#Help {Ata

140 messages · Page 1 of 1 (latest)

dire apex
#

there x many x'es (derivative)

WhatsApp_Image_2026-03-14_at_18.54.50.jpeg
supple ospreyBOT
glad heath
#

Sorry?

loud scaffold
# dire apex there x many x'es (derivative)

suppose x = pi. how are you going to add x pi number of times?
basically, x + ... + x = x^2 only works for integer values, but differentiating considers the whole real line. why it fails exactly is a bit more complicated

dire apex
#

i wonder why it fails

dire apex
#

depending on derivative

#

1 has to be equal to 2 :D

#

but this doesnt make sense at all so i need help so bad

loud scaffold
#

ah ok. do you agree that the sum only works for integer values? (i.e. you can't sum something 0.5 times)

dire apex
#

yes

#

i do

#

so this kinda is an exception?

loud scaffold
dire apex
#

sorry for that english is not my primary language

#

are there a way to prove it fails?

loud scaffold
#

no worries at all :) i guess you could say that, because this 'function' is discontinuous, it is not differentiable

#

(not sure if you have learned this yet)

dire apex
#

yes i meant that =)

loud scaffold
# dire apex i do

coming back to this: let's create a continuous version of this representation, and see what happens

dire apex
#

okey

loud scaffold
#

hmm, small problem hmmcat i thought i could do this with one variable, but i will need two variables and partial derivatives, which is not easy to understand

dire apex
#

partial derivatives

loud scaffold
#

eh lets try it. id like you to consider the function g(x) = x*f(x). what is the derivative of this function?

dire apex
#

g'(x)=f(x)+xf'(x) ?

#

in english its even harder :D

#

i think

loud scaffold
#

haha sorry 😅 but its correct

dire apex
#

oh

#

really

#

haha

#

i thought it was incorrect

loud scaffold
#

now, when f(x)=x, what is g(x)?

dire apex
#

hmm

#

if f(x)=x

#

then x'=1

#

so

#

g(x)=x?

loud scaffold
#

remember g(x)=x*f(x)

#

not g'(x)

dire apex
#

ohh

#

i dont know :(

loud scaffold
#

replace f(x) with x in the equation above

dire apex
#

x'?

#

g(x)=x'?

#

wait

#

noo

#

g(x)=x'.x

#

hmm

loud scaffold
#

just x*x, aka x^2

dire apex
#

oooh

loud scaffold
#

then from before, g'(x) = f(x) + xf'(x)

dire apex
#

yes

loud scaffold
#

if f(x)=x, then f'(x) = ?

dire apex
#

do we know the power of x?

#

if its just x

#

then 1

loud scaffold
#

yeah, no tricks here

#

just x^1

#

so g'(x) = ?

dire apex
#

haha

#

i didnt understand

loud scaffold
#

we have g'(x) = f(x) + x*f'(x)

dire apex
#

yeah

loud scaffold
#

now replace f(x) with x, and f'(x) with 1

dire apex
loud scaffold
#

x + x*1

dire apex
#

idk how to do star

dire apex
#

mb

#

x + x*1

loud scaffold
dire apex
#

what does star mean?

#

square?

loud scaffold
#

multiplication

#

2*3 = 6

dire apex
#

ooh

#

this makes sense

loud scaffold
#

so g'(x) = x + x = 2x

#

this is not very surprising

dire apex
#

lol

#

soo

loud scaffold
#

but here is the idea: we will pick a specific x, like x=1, and set f(x) = 1

#

so now, g(x) = x * 1

dire apex
#

yeah

#

yes

loud scaffold
#

what is g'(x)?

dire apex
#

g'(x) = 1

loud scaffold
#

yes

#

but here is the problem

dire apex
#

this wasnt suprising btw 😎

#

whats the problem?

loud scaffold
#

g'(x) represents the slope of y=x^2 at x=1

#

what is this slope?

dire apex
#

dont blame me but i dont know what ^ and slope means...

loud scaffold
#

let me rewrite

#

$g'(x)$ represents the derivative of $y=x^2$ at $x=1$ \
what is the derivative of $x^2$ at $x=1$?

analog turtleBOT
#

haseeb ♥

367809657273712641.png
dire apex
#

and 1 is 0

loud scaffold
#

the derivative is 2x, now if x=1 then 2x =

dire apex
#

x to the power of ''n'' the derivative is n*x to the power of n-1

dire apex
loud scaffold
#

yes

#

so g'(x) should be 2, but it is 1

dire apex
#

OOOOOH

#

g'(x) is a liar

loud scaffold
#

lol sure

dire apex
#

lol

#

so we just proved it fails right?

#

didnt we?

loud scaffold
#

yes, the formula we got doesn't hold under differentiation, in a sense

#

this is because we are considering a discontinuous function

#

we are only looking at x^2 at integer values, not as a whole line

dire apex
#

yes i understand

#

thanks

loud scaffold
#

i.e. our 'function' looks like this

image.png
#

which is not very differentiable

dire apex
#

its

#

not a function

#

also

#

its not i dont know what it means in english

#

we cant derivative on it

loud scaffold
#

there is no f'(x) for this f(x)

dire apex
#

this makes sense

#

YES

#

i wanted to say this

loud scaffold
#

=> not differentiable

dire apex
#

we cant have f'(x)

dire apex
#

tysm <3

#

even though i felt kinda insulted when u said it wasnt suprising :D

#

it helped a lot

#

i really appreciete it

dire apex
loud scaffold
#

but perhaps you and i have different ideas of surprising, and that is okay

dire apex