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.

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.
