What Are the Derivatives of |x| and [x]?

In summary, the conversation discusses the differentiability of the absolute value function and the greatest integer function. It is determined that |x| is differentiable everywhere except for at 0, where it has a cusp and is not differentiable. The greatest integer function is not continuous at integral values and therefore not differentiable at those points. The derivative of |x| is 1 for x>0 and -1 for x<0, which can be written as x/|x| for x ≠ 0.
  • #1
jobsism
117
0
Are |x| and [x] differentiable anywhere? If so, what're their derivatives?
 
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  • #2
Um...just about everywhere.
Take f(x) = |x| for 0 < x < ∞. Now, what's its derivative? (you should be able to figure this out)
Is this a homework question?
 
  • #3
[x], in particular, (I assume you mean either the "floor" or "ceiling" function) is constant for any non-integer x.
 
  • #4
I understand that the graph of |x| has got a cusp at 0, so that you can't draw a tangent there, and so itsn't differentiable at 0. I also understand that [x] (I meant the greatest integer function)isn't continuous at integral values, and so it isn't differentiable for integer values of x.

What I don't understand is the following statement made by one of my online friends: " gif is not diff at terminal points and modulus function at zero, but at other points gif gives 0 and mod gives mod x divided by x". Can anyone make sense of this? "gif" means greatest integer function.
 
  • #5
As Number Nine asked, what is the derivative of |x| for x>0? For x<0?
 
  • #6
The gif is constant between integers, so its derivative is 0 there. At integers it jumps, so there is no derivative.

|x| has derivative 1 for x > 0 and -1 for x < 0, this can be written as x/|x| for x ≠ 0.
 
  • #7
Oh, I'm sorry I missed that question. For x>0, derivative of |x| is 1, and for x<0, derivative of |x| is -1.

mathman said:
This can be written as x/|x| for x ≠ 0.

Now I get it! Thanks everyone!
 

What is the difference between |x| and [x] differentiability?

|x| differentiability refers to the differentiability of the absolute value function, where the derivative exists at all points except for 0. [x] differentiability refers to the differentiability of the greatest integer function, where the derivative exists at all points except for integers.

What is the meaning of differentiability?

Differentiability is a property of a function where the derivative exists at every point in the domain. This means that the function is smooth and has a well-defined slope at every point.

How can you determine if a function is differentiable at a specific point?

A function is differentiable at a specific point if the left-hand and right-hand derivatives at that point are equal. This means that the function is continuous and has a well-defined slope at that point.

What are the implications of a function being differentiable?

If a function is differentiable, it means that it is continuous and has a well-defined slope at every point in its domain. This allows us to use calculus to analyze the function and make predictions about its behavior.

Are |x| and [x] both continuous functions?

Yes, both |x| and [x] are continuous functions. This means that they have no breaks or gaps in their graphs and they can be drawn without lifting the pen from the paper.

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