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 ?
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.
Thank you guys for your help! Just to clarify this once more, is ?
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.
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:
the definition of f being differentiable at a point is that f ' exists at that pointBut couldn't there be a case, where the function f' isn't defined at some point, but f is still differentiable there?