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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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    @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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    @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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    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