Why is f(x,y) not differentiable at (0,0)?

In summary, the function f(x,y) = sqrt(abs(xy)) is not differentiable at (0,0) as shown by the limit definition of Df(x,y) where the limit along the 45 degree line in the first quadrant is 1, contradicting the existence of a zero map. This also proves that f(x,y) is continuous at 0.
  • #1
wakko101
68
0
Hello,

My question is as follows: Show that the function f(x,y) = sqrt(abs(xy)) is not differentiable at (0,0).

I was going to go with trying to show that the directional derivatives don't all exist here, but that would require finding the gradient, and I always get confused when trying to take the derivative of an absolute value. Essentially, this means that for xy larger than 0, f = sqrt(xy) and for xy smaller than 0, f = sqrt(-xy). But, of course, you can't have the square root of a negative number, so I'm confused...what should I do?

Thanks,
W.
 
Physics news on Phys.org
  • #2
Show that's it's not continuous?
 
  • #3
Try taking limits and show that the limit doesn't exist at (0,0).
 
  • #4
but which limits? If I want to use the definition of a derivative for multi variables, then I still need the gradient, don't I?
 
  • #5
Use the limit definition of Df(x,y). In this case, if f were differentiable at 0, then Df(0,0) would be the zero map. On the other hand, by approaching zero along the 45 degree line in the first quadrant one would then have the limit to be 0 in spite of the fact that the limit is clearly 1. You'll have to try it to see what I mean.
 
  • #6
Since this problem is old now, I will give out my full solution now:

http://img113.imageshack.us/img113/5482/mysolutiontf9.jpg [Broken]

JFonseka said:
Show that's it's not continuous?
Actually, f(x,y) is continuous at 0:

http://img404.imageshack.us/img404/1144/continuityproofqf0.jpg [Broken]
 
Last edited by a moderator:

What is the derivative of the absolute value function?

The derivative of the absolute value function, denoted as f(x) = |x|, is not defined at x = 0. However, for all other values of x, the derivative is equal to 1 or -1, depending on the sign of x.

How do you find the derivative of an absolute value function?

To find the derivative of an absolute value function, you can use the definition of derivative, which is the limit of the difference quotient as h approaches 0. This can be simplified using the properties of absolute value and the limit laws.

Why is the derivative of absolute value not defined at x = 0?

The derivative of absolute value is not defined at x = 0 because the function is not differentiable at that point. This is because the slope of the function changes abruptly from -1 to 1 at x = 0, resulting in a vertical tangent line.

What is the graph of the derivative of absolute value?

The graph of the derivative of absolute value is a piecewise function with a horizontal line at y = -1 for x < 0 and a horizontal line at y = 1 for x > 0. It has a vertical tangent line at x = 0.

What are some real-world applications of the derivative of absolute value?

The derivative of absolute value can be used in modeling situations where there is a sudden change in direction or velocity, such as in physics and engineering. It can also be used in optimization problems, where the absolute value function represents a constraint in the problem.

Similar threads

  • Calculus and Beyond Homework Help
Replies
6
Views
505
  • Calculus and Beyond Homework Help
Replies
2
Views
538
Replies
1
Views
1K
  • Calculus and Beyond Homework Help
Replies
4
Views
809
  • Calculus and Beyond Homework Help
Replies
9
Views
4K
  • Calculus and Beyond Homework Help
Replies
15
Views
2K
  • Calculus and Beyond Homework Help
Replies
2
Views
459
  • Calculus and Beyond Homework Help
Replies
4
Views
9K
  • Calculus and Beyond Homework Help
Replies
5
Views
1K
  • Calculus and Beyond Homework Help
Replies
11
Views
2K
Back
Top