Uniqueness of the derivative of a function $f : \mathbb{R} \to \mathbb{R}$ 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.

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.
