# Diffrentiability of function with two variable

• The Rahul
In summary, The Rahul found a way to solve a problem that involves finding the derivative of a function at a particular point, but found an easier way to do it using x=y3.
The Rahul

Welcome to PF!

Hi The Rahul! Welcome to PF!

Show us one or two examples that you've had difficulty with, together with your attempts to solve them.

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!

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)

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

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

tiny-tim said:
Hi The Rahul!

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)

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

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

Ah, yes! Let sleeping mentors lie! We are very grouchy until we have had our coffee. (Except for evo, of course.)

## What is the definition of differentiability for a function with two variables?

Differentiability for a function with two variables means that the function has a well-defined tangent plane at every point in its domain. This means that the function is smooth and has no sharp turns or corners.

## What is the difference between partial derivatives and total derivatives?

Partial derivatives measure the rate of change of a function with respect to one variable while holding all other variables constant. Total derivatives, on the other hand, measure the overall rate of change of a function with respect to all its variables.

## How do you determine if a function with two variables is differentiable at a specific point?

A function with two variables is differentiable at a specific point if all its partial derivatives exist and are continuous at that point. This means that the function is smooth and has a well-defined tangent plane at that point.

## What is the relationship between differentiability and continuity for a function with two variables?

A function with two variables can be differentiable without being continuous, but it cannot be continuous without being differentiable. This means that a function must be continuous in order to be differentiable, but being differentiable does not necessarily guarantee continuity.

## How can differentiability be used to determine the behavior of a function with two variables?

Differentiability can be used to determine the local behavior of a function with two variables, such as the existence of maxima and minima. It can also be used to approximate the value of a function at a specific point using its tangent plane.

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