Diffrentiability of function with two variable

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Discussion Overview

The discussion revolves around the differentiability of functions with two variables, specifically focusing on examples where differentiability fails at certain points. Participants explore methods to demonstrate non-differentiability and share their approaches to solving related problems.

Discussion Character

  • Homework-related
  • Technical explanation
  • Exploratory

Main Points Raised

  • One participant, The Rahul, seeks help with differentiability problems, specifically mentioning the function f(x,y) = |xy|^{1/2} and its non-differentiability at (0,0).
  • The Rahul also references another function, f(x,y) = xy / √(x^2 + y^2), noting that the method in his book differs from his approach.
  • Another participant suggests that to prove non-differentiability at (0,0), it suffices to find one curve along which the function is not differentiable, recommending the curve x = y^3 for further investigation.
  • There is a mention of the derivative existing along straight lines at (0,0), indicating that further analysis is needed beyond linear approaches.
  • Participants discuss the appropriateness of certain curves for checking differentiability, with one noting that x = y^2 may complicate the analysis due to square roots.

Areas of Agreement / Disagreement

Participants express differing methods for proving non-differentiability, with no consensus reached on a single approach. The discussion remains exploratory, with multiple viewpoints on how to analyze the functions presented.

Contextual Notes

Some assumptions about the functions and their behavior near (0,0) are not fully explored, and the discussion does not resolve the mathematical steps involved in proving differentiability or non-differentiability.

The Rahul
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Hii please help me to Solve problems diffrentiability of function with two variable with one method.
 
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Welcome to PF!

Hi The Rahul! Welcome to PF! :smile:

Show us one or two examples that you've had difficulty with, together with your attempts to solve them. :wink:
 
The Rahul, you just sent your reply to the mentor's forum. I'm reposting it here for you. To reply, please use either the "quote" button on the bottom right, or "New Reply", not "report".

The Rahul said:
Hii Tiny tim,thnx for ur warm welcome,
Now I continue my problem=
In this Ques.—f(x,y)={|xy|}^1/2 Not differentiable at (0,0)
I find out Both Partial derivative fx and fy and solve the ques.
But in that ques.== f(x,y)= xy /√(x^2+y^2)
in my book method is diffrent.
in that ques method is to find according to y=mx and x=y^3.
so Plz help me...:-D
 


Hii Tiny tim,thnx for ur warm welcome,
Now I continue my problem=
In this Ques.—f(x,y)={|xy|}1/2 Not differentiable at (0,0)
I find out Both Partial derivative fx and fy and solve the ques.
But in that ques.== f(x,y)= xy /√(x2+y2)
in my book method is diffrent.
in that ques method is to find according to y=mx and x=y3.
so Plz help me...:-D
 
Hi The Rahul! :smile:

To prove that a function f(x,y) is not differentiable at (0,0),

we only need to find one curve along which it is not differentiable (and then we can stop).

If f(x,y) = xy /√(x2+y2) = xy/r,

then the derivative along any straight line does exist at (0,0), so we can't stop there, we need to check other ways of approaching the origin …

in this case, the easiest curve to check is x=y3 (or y=x3) :wink:

(x=y2 is awkward, because it gives you awkward square-roots​


btw, since you've woken the mentors :wink:, please note that txt-english (eg "please", "ur") is against the forum rules! :bugeye:
 
tiny-tim said:
Hi The Rahul! :smile:

To prove that a function f(x,y) is not differentiable at (0,0),

we only need to find one curve along which it is not differentiable (and then we can stop).

If f(x,y) = xy /√(x2+y2) = xy/r,

then the derivative along any straight line does exist at (0,0), so we can't stop there, we need to check other ways of approaching the origin …

in this case, the easiest curve to check is x=y3 (or y=x3) :wink:

(x=y2 is awkward, because it gives you awkward square-roots​


btw, since you've woken the mentors :wink:, please note that txt-english (eg "please", "ur") is against the forum rules! :bugeye:



Thank you for your precious reply.It works.:smile:
 
Ah, yes! Let sleeping mentors lie! We are very grouchy until we have had our coffee. (Except for evo, of course.)
 

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