The meaning of f’(a) [read “F Prime”] is the slope (tangent) at point a.
There are three main ways of thinking of the derivative.
It is the slope of a line
It is the Instantaneous velocity
It is the rate of change
Some Basic Derivative problems
(Practice and be able to explain these, using the Derivative formula (Limit h approaching 0…)
Find the Tangent line at point x = 2
y minus initial y equals the slope times x minus initial x.
Domain in f(x) and f’(x)
The Domains of a function and it’s derivative are not always equal to each other, the Domain of f(x) is greater than or equal to the domain of f’(x). We say that f is differentiable in f if f’(x) exists. To prove f’(x) exists, the following three must exist.
We say that f is differentiable if f’(a) exists for all a in the domain.
The smooth graph is differentiable. A graph with corners is not. Eq 4, above, is an example of a graph which is not differentiable at 0. To be differentiable implies continuity.
A function is continuous if the following two limits exist, and when multiplied by each other equal something.
Meaning of Derivative
The Derivative is the line of the slope. When it crosses an axis you have reached a minimum or maximum value. The line of negative slope is going down. The line of positive slope is increasing. Look at Tangent lines, they approximate the graph and tell the direction up or down.
f’(x) is negative, x is decreasing near x.
f’(x) is positive, x is increasing near x.