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