ALL ABOUT THE DERIVATIVE
~ The derivative of a function is developed in the early stages of
Calculus. It serves as the foundation of one of the two major operations
of calculus, namely differentiation. The 2nd major operation of calculus is integration which is developed later.
~ It is called the derivative since it is derived from a parent
function. It will give very important information
of properties of the parent function that lead to the solution to
many problems in science & business applications.
~ Some of those properties include the following: The rate of
change of the dependent variable of the parent function with respect to
the independent variable. For example, if the equation of the
parent function takes the form R=f(x), where R is the revenue
collected when selling x units of an item, the derivative will tell us
how revenue changes at any level, x, of sales. Another popular
way to phrase the problem would be, "how fast is the revenue changing
when the sales level, x, is at a certain value"?
~ In the above example, the units for the derivative could be in dollars per item for a specific
number of items. When rate of change is mention, it assumes an
instantaneous rate. Average rate of change requires no knowledge
of calculus and is simply the slope of the line connecting two
points on a function. The word Average will be use, if this is
what we want, otherwise, it will be assumed that the rate of change is
instantaneous. This will require getting the derivative.
~ The derivative also gives us the slope of any tangent line drawn to
the parent function at a specified point. Unlike the average rate of
change between two points (the slope of a
line connecting these points), the derivative gives the slope at one
point. This will serve as a description of behavior of the parent
function at this isolated point (there lies the power of the
derivative).
~ Different points on the parent function have different sloped tangent
lines (assuming a nonlinear parent function) so the derivative will
give us a way of analyzing the parent function at any isolated point
desired.
~ A linear parent function will have a derivative which is the same at
all points on that curve since the slope between any two points, no
matter where you take them, is always the same. So, in this case,
the average rate of change between any two points = the rate of change
at any point. The slope of a linear function is constant.
For nonlinear functions, we need the derivative to find the
slope at different points since the slopes of the tangent lines drawn
at these points change as we travel along the curve.
~ In business applications, the derivative gives a marginal value at an
indicated value of x (items produced or sold). So, to find
the marginal revenue at a sales level x =20, you would substitute x=20
into the derivative of the revenue function, which would be given. Marginal
values usually deal with revenue, cost, & profit tied to the parent function used in the specific problem
and the in-dept study of this is called Marginal Analysis (covered
in a basic business calculus course).
~ So, in summary, the derivative is usually interpreted in 4 ways:
1) rate of change of one quantity (dependent variable) with respect to another (independent variable).
2) the slope of tangent lines to the graph of the parent function at various places.
3) for business applications, it will give marginal values of the quantity examined
4) for distance (s) - time (t) functions, the derivative gives
velocity (v) since it measures how distance changes wrt time (i.e.,
ft/sec or miles/hr). Please note that velocity is directed speed.
Speed is a scalar quantity (magnitude only) while, with a direction, gives
velocity, a vector quantity.