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There are several equivalent ways of defining a function. We know that a differentiable function $f : \mathbb{R} \to \mathbb{R}$ is uniquely defined when its values are specified at every point in $\mathbb{R}$. Now the question is : Is the derivative of such a function $f$ always unique ?

PS: Pardon me if its a very trivial question !

EDIT 1: the definition of the derivative is same as usual...i mean that given in the answer by Jonas Meyer and so is the definition of differentiability.

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    @PEV : please let me know what have you edited in the question.2011-01-20
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    at the bottom of the question, there is now a box showing that PEV edited the the question some time ago. You can click on the time there (now it reads 3 mins ago, so just click on the words "3 mins"). This will lead you to the revision history page, which shows that PEV removed the emphasis you placed on the last sentence of the first paragraph.2011-01-20
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    @Willie Wong : that was a subtle edit.Thanks for pointing out. I couldn't notice it the first time i saw before asking.2011-01-20
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    in view of the answer given by PEV below, perhaps you need to clarify whether you are asking for two functions $f,g$ such that their derivatives coincide, or for a function $f$ such that it has "two distinct derivatives". In the latter case you should also update the question with your definition of "differentiable function" and "derivative".2011-01-20
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    @Willie Wong : Ok. give me a minute.thanks2011-01-20

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Given $f:\mathbb{R}\to\mathbb{R}$, the derivative of $f$ at $x\in\mathbb{R}$, if it exists, is typically defined to be $f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$. Real limits are unique when they exist, so this unambiguously assigns (at most) one value to $f'(x)$. Therefore the derivative is unique. Assuming $f$ is everywhere differentiable, this means that $f':\mathbb{R}\to\mathbb{R}$ is a ("well-defined") function.

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    @Jonas Meyer : I am working with a sequence of functions which are converging uniformly and also their derivatives.But somehow the derivative of the limit function for all the functions in the sequence are not unique.the derivatives converge uniformly though.2011-01-20
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    @Rajesh: That is not possible. If the sequence of derivatives converges uniformly, then it converges to the derivative of the limit of the original sequence. Both limits of sequences of real valued functions (when they exist) and derivatives (when they exist) are unique. (More generally, limits in a Hausdorff topological space are unique.)2011-01-20
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    @Rajesh: can you exhibit that sequence of functions? What you wrote contradicts a well-known theorem (see http://en.wikipedia.org/wiki/Uniform_convergence#To_differentiability for a statement) that can be found in e.g. Rudin's _Principles of Mathematical Analysis_.2011-01-20
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    @Willie Wong , @Jonas Meyer : After re-checking my derivations, I was thinking of the same thing this morning (while sitting in a class) as what you said. I want to pose this question in a different context (a different definition of derivative)....Could i edit the same question or should I write it in a new question ?2011-01-21
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    @Rajesh: It sounds like it might be a very different question, so it would probably be easiest to ask it as a new question.2011-01-21
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    @Rajesh: I agree with Jonas. You probably should pose it as a new question.2011-01-21
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The answer is, if it exists the derivate is unique (unlike a primitive). Even a more abstract form of the derivate, the weak derivate is unique ( http://en.wikipedia.org/wiki/Weak_derivative ). A reason for this is the uniqueness of limits, as the derivate is basically a limit.

However the converse is not true, a derivate doesn't have a unique function associated with it, see the example PEV gave you.

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    @user3123: it'd be better to say *primitive* instead of *integral* here.2011-01-20
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    @lhf thank you my english is not too good.2011-01-20
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    IMO, "primitive" sounds a little old-fashioned (although there's nothing explicitly wrong with that). Most people today would say "anti-derivative", I think.2011-01-20