Why Is the Absolute Value Function Not Differentiable at x=0?

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In summary, the conversation discusses the differentiability of |x| at x=0 and the reasons behind it. The graph of |x| is not differentiable at x=0 because it does not have a unique tangent at that point. This is due to the limits from the left and right not being equal, resulting in an abrupt change in the tangent line. The subgradient at x=0 is not a singleton, and the lack of a continuous change is the reason why |x| is not differentiable at x=0.
  • #1
welatiger
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0
Hello;

I want to know why |x| is not differentiable at x=0.

Thanks
 
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  • #2
The limits from the left and from the right are not equal.
 
  • #3
welatiger said:
Hello;

I want to know why |x| is not differentiable at x=0.

Thanks

Just look at the graph of |x|. Does it have a unique tangent at x=0?
 
  • #4
symbolipoint said:
The limits from the left and from the right are not equal.

.. What is this..

Ray Vickson said:
Just look at the graph of |x|.

Aaah, that's a physicist's answer!
 
  • #5
Blast those physicists! (Especially here where that probably the best answer.)
 
  • #6
Compare [tex] lim_{h \to 0^{+}} \frac{|x+h|-|x|}{h} [/tex] and [tex] lim_{h \to 0^{-}} \frac{|x+h|-|x|}{h} [/tex]
 
  • #7
jeppetrost said:
.. What is this..



Aaah, that's a physicist's answer!

More "mathematical": the subgradient is not a singleton at x = 0.
 
  • #8
symbolipoint said:
The limits from the left and from the right are not equal.

I should not have said that; it is incorrect.

Actually, the change in tangent line makes an abrupt, not continuous change at x=0. The lack of a continuous change is the reason why the absolute value of x is not differentiable there.
 

FAQ: Why Is the Absolute Value Function Not Differentiable at x=0?

What is the meaning of the term "differentiation" in the context of abs(x)?

The term "differentiation" refers to the process of finding the rate of change or slope of a function at a specific point. In the case of abs(x), it involves determining the rate of change of the absolute value function at a given value of x.

How is the differentiation of abs(x) different from other types of differentiation?

The differentiation of abs(x) is unique because the absolute value function is not differentiable at x = 0. This means that the slope of the function at this point is undefined, making it unlike most other functions which are differentiable at all points.

What is the derivative of abs(x)?

The derivative of abs(x) is a piecewise function, where the derivative is equal to 1 for all positive values of x and -1 for all negative values of x. At x = 0, the derivative does not exist.

How is the derivative of abs(x) used in real-world applications?

The derivative of abs(x) is used in various real-world applications, such as in physics to calculate velocity and acceleration, in economics to determine marginal cost and revenue, and in engineering to optimize systems and processes.

Can the differentiation of abs(x) be generalized to other absolute value functions?

Yes, the process of differentiation can be applied to any absolute value function, not just abs(x). The same rules and concepts apply, such as the derivative being undefined at the point where the absolute value function crosses the x-axis.

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