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.