Derivative of Absolute Value Confusion

In summary, the conversation is about a discrepancy in the derivative of an absolute value function. The person asking the question believes the derivative should be x/abs(x), while the book has abs(x)/x. The expert explains that both derivatives are equal, but the problem may arise when the function is not differentiable at 0. The conversation ends with the person realizing it may have been a typo or misinterpretation.
  • #1
russ_watters
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Quick question (hopefully): I'm helping my girlfriend with some calculus homework over the phone and it looks like the book has a different answer for the derivative of an absolute value than I'm seeing in other places. I'm seeing it as (and I derived it myself to be) x/abs(x) whereas the book (apparently) has abs(x)/x. But it seems to me that these should be equal. Are they?

Why this matters is if you try to apply this to problems, you get some very ugly things.

What am I missing?
 
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  • #2
x2=|x|2
x*x = |x|*|x|
x/|x| = |x|/x

So yes these things are equal. The ugly things might be coming up because the function is not differentiable at 0 and of course neither of these is defined at zero. If your problem involves crossing x=0 weird things can happen. Can you give an example?
 
  • #3
Given: y=|x|+1/x
Solution: y=(x2)1/2+x-1
y'=1/2(x2)-1/22x-x-2
y'=x/|x|-1/x2
Can also be written: y'= (x|x|-1)/x2

But unless I'm mishearing her, the solutions manual says:
y'=|x|/(x-1/x2)
...and they don't look equal to me.
 
  • #4
Maybe it's just a typo (or a mishear), intended to be y'=|x|/x - 1/x2.
 
  • #5
Yeah, I asked several times, but it is possible that it is either.
 
  • #6
Yeah, I got her on a webcam and it was an algebra comprehension issue. That's what I thought but I wanted to make sure I wasn't crazy.
 
  • #7
y = |x|
implies y = ax and y = -ax
therefore y' = a and y' = -a
 

What is the derivative of absolute value?

The derivative of absolute value is a mathematical concept that represents the rate of change of a function at any given point. It is defined as the slope of the tangent line to the absolute value function at that point.

How do you find the derivative of absolute value?

To find the derivative of absolute value, you can use the definition of the derivative as the limit of the difference quotient. This involves taking the limit as the change in input approaches 0. Alternatively, you can use the properties of absolute value and the chain rule to simplify and evaluate the derivative.

Is the derivative of absolute value defined at all points?

No, the derivative of absolute value is not defined at points where the function is not differentiable, such as at the point where the function changes direction (the sharp point). At these points, the left and right derivatives are not equal, so the derivative does not exist.

What is the derivative of the absolute value of x?

The derivative of the absolute value of x is equal to the sign of x, defined as -1 for negative x and +1 for positive x. This is because the derivative of the absolute value function is a piecewise function, with a slope of -1 for x < 0 and a slope of +1 for x > 0.

Can the derivative of absolute value be negative?

Yes, the derivative of absolute value can be negative. This occurs when the input to the absolute value function is negative, as the derivative at this point is -1. However, the derivative can also be positive or undefined, depending on the input value.

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