The Definition of the Derivative : In the previous section we saw that the calculation of the slope of a tangent line, the instantaneous rate of change of a function, and the instantaneous velocity of object at x =a all needed us to calculate the following limit.

We also saw that along with a small change of notation this limit could also be written like this,

This is such significant limit and it arises in so several places that we give it a name.  We call it a derivative.  Following is the official definition of the derivative.

Definition : The derivative of f ( x ) w.r.t. x is the function f ′ ( x ) and is described as,

Note as well that we replaced all the a's in (1)  along with x's to acknowledge the fact that the derivative is actually a function as well. We frequently "read" f ′ ( x ) as "f prime of x".