##### Difference Quotients & Definition of the Derivative

In the second posting on Local Linearity, we saw that what we were doing, finding the slope to a nearby point, looked like this symbolically:

This expression is called the *Forward Difference Quotient* (FDQ). It kind of assumes that *h* > 0.

There is also the *Backwards Difference Quotient* (BDQ):

The BDQ also kind of assumes that *h* > 0. If *h* < 0 then the FDQ becomes the BDQ and vice versa. So these are really the same thing. The limit (if it exists) as *h* approaches zero is the slope of the tangent line at whatever *x* is and this is important enough to have its own name. It is called the derivative of *f* at *x* with the notation (among others) :

Since *h* must approach 0 from both sides, this expression incorporates the FDQ and the BDQ in one expression.

To emphasize that *h* is a “change in *x*” this limit is often written