Finding out if a function can be derived at certain point

Hello,

One question that the professor of my calculus course likes to include in his exams is, if a certain function f(x) can be derived at a certain point (e.g. x=0).

In order to find this out, do I always have to find the limit or can I also just find the derivative and then look at the limit ?

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Originally Posted by thomasdotnet

Hello,

One question that the professor of my calculus course likes to include in his exams is, if a certain function f(x) can be derived at a certain point (e.g. x=0).

In order to find this out, do I always have to find the limit or can I also just find the derivative and then look at the limit ?

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The latter would seem OK. But you're best advised to go ask your professor - he's the one who will be marking your exam, not me.

You don't always have to use the limit

consider f(x) = x^(1/3)

then f ' (x) =1/3 x^(-2/3) = 1/[3x^(2/3)] so f is not differentiable at 0

Often you simply compute f ' and see where it does not exist.

In other cases like f(x) = |x|

then at 0 then f ' = 1 if x>0 and f ' = -1 x< 0

here you use lim f ' at 0 to see it is not differentiable at 0

hope this helps

By the way we talk about whether or not a function can be differentiated
at apoint not derived.

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Originally Posted by Calculus26

You don't always have to use the limit

consider f(x) = x^(1/3)

then f ' (x) =1/3 x^(-2/3) = 1/[3x^(2/3)] so f is not differentiable at 0

Often you simply compute f ' and see where it does not exist.

In other cases like f(x) = |x|

then at 0 then f ' = 1 if x>0 and f ' = -1 x< 0

here you use lim f ' at 0 to see it is not differentiable at 0

hope this helps

By the way we talk about whether or not a function can be differentiated
at apoint not derived.

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I think that last point depends upon whether you speak "British" English or "American" English!

What ? The British speak English ?

Thank you guys for your help! Just to clarify this once more, is ?

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Originally Posted by Calculus26

Often you simply compute f ' and see where it does not exist.

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But couldn't there be a case, where the function f' isn't defined at some point, but f is still differentiable there?

Regarding differentiable vs. derivable: Since the Americans vote for one and the British vote for the other, I guess we'll have to let the Australians decide, which one we should use. (Wink)

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http://www.mathhelpforum.com/math-he...5bab7600-1.gif

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What the above quote is saying is that f ' (x) not only exists but is continuous which is a more stringent condition.

In fact I was in advanced calculus before i even saw an example of a function which was differentiable at a point but the derivative was not continuous at that point.

Again you can think of functions which are defined at a point but are not

continuous for eg f(x) = |x|/x if x does not equal 0 and = 1 if x = 0

f(0) = 1 but f is not continuous at 0.

as to your statement:

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But couldn't there be a case, where the function f' isn't defined at some point, but f is still differentiable there?

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the definition of f being differentiable at a point is that f ' exists at that point

Thank you Calculus26 for that explanation, now it's a lot clearer to me.

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Originally Posted by Calculus26

Again you can think of functions which are defined at a point but are not

continuous for eg f(x) = |x|/x if x does not equal 0 and = 1 if x = 0

f(0) = 1 but f is not continuous at 0.

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Isn't undefined at x=0?