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Differentiating absolute values Watch

1. Find derivative of and at using the definition of the derivative.

For at :

After simplifying i got

Since

For at :

After simplifying i got

Now i know that the derivative of at does not exist, but the derivative of derivative of at is 0. Why doesn't what i have done work for both?
2. (Original post by John taylor)
Find derivative of and at using the definition of the derivative.

For at :

After simplifying i got

Since

For at :

After simplifying i got

Now i know that the derivative of at does not exist, but the derivative of derivative of at is 0. Why doesn't what i have done work for both?
Your mistake is in saying that - can you see what it should be?
3. (Original post by Mark13)
Your mistake is in saying that - can you see what it should be?
Why is it a mistake, isn't this true by definition and how come it works for the second one if its wrong?

And i guess you mean
4. (Original post by John taylor)
Why is it a mistake, isn't this true by definition and how come it works for the second one if its wrong?

And i guess you mean
The problem is, it isn't true. You're saying

But what if ?
5. (Original post by Noble.)
The problem is, it isn't true. You're saying

But what if ?
How come it works for the derivative of
6. (Original post by John taylor)
How come it works for the derivative of
Sometimes incorrect maths gives you the correct answer.
7. (Original post by John taylor)
After simplifying i got
This bit is wrong. Basically, the problem is apparent when you unpack the definition of what a limit really is. What fails is that there is no open set containing where the in your denominator is always positive (what you tacitly assume in making your erroneous conclusion.

The easiest way to expand upon this is to write:

and consider the one sided limits

Note that these differ, which means that the two sided limit

doesn't exist.
8. (Original post by Mark85)
This bit is wrong. Basically, the problem is apparent when you unpack the definition of what a limit really is. What fails is that there is no open set containing where the in your denominator is always positive (what you tacitly assume in making your erroneous conclusion.

The easiest way to expand upon this is to write:

and consider the one sided limits

Note that these differ, which means that the two sided limit

doesn't exist.
For the derivative of would i need to still consider left and right limit of 0, even though it exists.
9. (Original post by John taylor)
For the derivative of would i need to still consider left and right limit of 0, even though it exists.
I assume you mean and that is how you conclude the limit exists.

is differentiable at and if and only if has both left- and right-derivatives at and
10. (Original post by Noble.)
I assume you mean and that is how you conclude the limit exists.

is differentiable at and if and only if has both left- and right-derivatives at and
So for i got

so the the derivative exists and is 0 at x = 0, since both the left and right limit are the same. Is this right?
11. (Original post by John taylor)
So for i got

so the the derivative exists and is 0 at x = 0, since both the left and right limit are the same. Is this right?
Yes.
12. (Original post by John taylor)
For the derivative of would i need to still consider left and right limit of 0, even though it exists.
Although what Noble writes above is true and answers the question you were asking I should add:

One sided limits are a tool... you don't need them to define a general limit. What Noble posted is a useful theorem; not a definition. I mean, go back and look at the actual definition of a limit - no one sided limits are mentioned.

The point is, for certain functions, it is useful to consider limiting behaviour when we think of our variable approaching some point, say, seperately from the left and right. For example, if the function is defined piecewise

e.g.

then we can write down simple formulae when dealing with the right and left one sided limits in terms of and .

In general, when considering the (two sided) limit at , we have to consider open intervals containing so we can't just pick or and base our calculations off that. That was essentially the mistake you made in your erroneous example.

The take home message here is: go read, learn and inwardly digest the definition of a limit else you are wasting your time 'doing' such examples and asking questions about them.

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Updated: March 12, 2013
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