Can someone help me prove this derivative by definition?

[silk wagon]

differentiable means that the function's d erivative exists

but if the derivative is 0 its not differentiable

if its continous, then the left hand limit is equal to the right hand limit

[subtle pulsar]

Well yeah, but it has a derivative right? It's just that at x = 1 the derivative does not exist, or does that mean that the derivative in general does not exist?

[subtle pulsar]

I see, yeah, but is that enough for the proof?

[silk wagon]

if the derivative is 0 its not dfiferentiable at that point

[silk wagon]

just prove that from both sides its equal

[subtle pulsar]

Yeah, but to prove that, don't I need to find the derivative and show it's 0 at x = 1?

[silk wagon]

yeah

just do that

you know the rule for the derivative of modulus of x

right

its x for larger than 0 and -x for less than 0

[subtle pulsar]

But it says by definition, so I think I need to do it rigorous as well

[subtle pulsar]

Yeah, I saw that in high school, but it hasn't been explained in uni so far. Guess the same still applies then

[silk wagon]

yeah

[silk wagon]

obviously, you need to show the proof

[subtle pulsar]

is the derivative of |x| just x/|x|?

[silk wagon]

yeah

but basically, for this |x| its

|x-1|*

its x-1 for x larger than 0

and 1-x for less than 0

[subtle pulsar]

Yeah, so I need to find two derivatives; one for x >0 and one for x < 0

[silk wagon]

no

since 1 is larger than 0

just plug in 1 into the x-1 formula

which is 0

[subtle pulsar]

Ah yeah, I forgot about the 1 for a sec. So larger than 0

[silk wagon]

listen basically

for continuity just show that left hand and right hand limit are equal to each other

and then for differentiability

just show that the dreivative is 0

for it being continuous, itj ust means the function is defined at a point

[subtle pulsar]

Yeah, but it's hard when your new to proof-based math, so Ill try to find the derivative using the rigorous way first

[silk wagon]

so the limit from the left and lmit from right is equal

[silk wagon]

yeah thats how you're supposed to do it i guess

wait with derivative principles?

[subtle pulsar]

Yeah limit of h -> 0

I mean, that's the way the exercise is phrased

[silk wagon]

yeah sure

if you want to go ahead

[subtle pulsar]

I'll first do the continuous thing though, is that just x -> 1- and x -> 1 for the equation? That way both sides equal to 1, so that proves continuity correct?

[silk wagon]

yeah

just prove that

and you're good

actually i think its saying continuous in general

or is it saying continous at x = 1

[subtle pulsar]

''Prove by definition that f is continuous, but not differentiable at x = 1.''

But i proved it with the limits, so that should be good

[silk wagon]

ah

okay

[silk wagon]

good

[subtle pulsar]

Now the hardest part though lol

Do you have any idea how I should go about rigorously defining the derivative?

[silk wagon]

f'(1) = lim h goes to 0 of f(1+h) - f(1) all divided by h

[subtle pulsar]

but f(1) is just 1, are you sure I need to proceed with that?

[silk wagon]

yeah

[subtle pulsar]

so then I get ((1+h) - 1)/h correct?

[silk wagon]

wait what

n

o

you know definitin of derviative right

[subtle pulsar]

I have seen it yeah, but this is like the second time I am using it lol

[silk wagon]

$[ \lim_{h\to0} \frac{f(1+h) - f(1)}{h}$

[warm mason]

igotaclue
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[silk wagon]

this is the defnition of the derivative for f'(1)

plug in 1+h into f(x)

and 1 into f(x)

[subtle pulsar]

Yeah, because we plugged 1 in for x

[silk wagon]

and then 1+h in for x

and then divide all fo that by h

[subtle pulsar]

f(1+h) = (1+h)^2 - 2 |(1+h) - 1| right?

[subtle pulsar]

Or can I just remove abs?

[silk wagon]

yeah it would be h^2 + 1 all divided by h

so the derivaitve doesnt exist at thep oint x = 1

because the deriivative is infitiy

a vertical tangent

its not differentiable if the derivative is either 0 or it doesnt exist

[subtle pulsar]

Just for my understanding; we get h^2 + 1 . that is h + 1/h, so if h goes to 0 zero the outcome would be undefined correct?

[silk wagon]

as h goes to 0, it would be infinity

or undefined yes

it does not exist

so the derivative does not exist, which proves that f(x) is not differentiable at x = 1

if you dont understand i can make it more concise

[subtle pulsar]

Thank God, we found it. I understand it much better now, so thank you so much for this

[subtle pulsar]

although, I have to say that the whole rigorous derivative is still difficult for me, but it at least makes some sense now

[silk wagon]

basically its just simplifing

if you need anyhelp you can dm about anything

this is uni? what course is it?

[subtle pulsar]

Calculus 1, but it has some analysis too (cauchy sequence, epsilon-delta those things)

[silk wagon]

oh god, epsilon delta is annoying as hell

cauchy is when sequence progresses and terms get closer and closer i belive

not sure though

[subtle pulsar]

wow, that would be super helpful as I still need to solve some other questions, although I hope those will be a bit easier for me, so I won't be too much trouble asking all kind of things lol

[silk wagon]

nah its fine, im jobless anyways. Still got literature hw to do though

[subtle pulsar]

Yeah, basically as the sequence goes on things become arbitrarily closer to each other

[silk wagon]

if you need help, pop in my dms, its cool

[silk wagon]

yeah ive heard that

[subtle pulsar]

Cool, thank you so much!!

[silk wagon]

just asking, could you send me the pdf

i just want to see. Which university do you go to?

[subtle pulsar]

Which pdf?

[silk wagon]

for your problem set

[subtle pulsar]

If you want to know its best we move to dms I guess, as it is not really related to the question anymore (not sure how strict they are here)

[silk wagon]

yeah sure

if your doubt is cleared you may want to close the psot

[silk wagon]

$[ \lim_{h\to0} \frac{(h+1)^2 - 2|(h+1)-1|}{h}$

[warm mason]

igotaclue
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[silk wagon]

$\lim_{h\to0} \frac{h^2 + 1}{h}$, which is then equal to $\lim_{h\to0} h + \frac{1}{h}$ which as we know is undefined.$

[warm mason]

igotaclue
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[subtle pulsar]

.close