Last edited 22-Sep-09
DERIVATIVES
This section of
abstractmath.org is devoted solely to
understanding the concept of derivative by showing you graphs of curves and their derivatives. The idea is for you to stare at them and see
visually how the function and its derivatives are related to each other. You
don’t have to know the formula for the derivative to understand this. Of course, you need
the formula if you are going to use the derivative.
One of the values of putting
things on the web instead of in a book is that I can include many more examples
than would be practical to put in a book.
This
section is sketchy and will eventually be expanded. For more details about derivatives see here.
Derivatives
A continuous function 
,
where S is a subset
of ,
may have a derivative, which is another function . Thus
taking the derivative maps functions to functions. Such mappings are examples of operators.
Definition
Of course, the curve may not have a tangent line at (example). In that case the value is undefined.
This is a conceptual definition
of “derivative”. The “conceptual
definition” isn’t a mathematical definition until you define the tangent
line! The definition of “slope of the
tangent line” requires limits. If you work out all the definitions
concerned, you get an epsilon-delta
definition of derivative that allows you to prove statements about them.
Computing derivatives
In a first calculus course you learn methods for computing derivatives, such as
¨ Formulas for polynomial, trig, exponential and log (MW, Wi) functions.
¨ Methods for reduction, formulas that reduce
the derivative of one function to a formula involving derivatives of other
functions (which you hope are simpler): Sum and product rules, chain rule, quotient rule.
Applications
You also may learn many applications
of derivatives:
¨ derivative is rate of change,
¨ derivative of velocity is acceleration,
¨ second derivative shows concavity,
¨ when the derivative is zero there might be a maximum or minimum,
… and so on.
Terminology
The second derivative
is the derivative of ,
the third derivative is the derivative of ,
and in general, the (n+1)st derivative
is the derivative of the nth
derivative (example).
The nth derivative of f may also be denoted by ,
so that ,
and so on.
Graphs of functions and their derivatives
The graphs below show (part of) the graph of a function f(x)
along with one or more of its derivatives.
In each case, the graphs are colored according to this scheme:
|
Function
|
blue
|
|
First derivative
|
red
|
|
Second derivative
|
green
|
|
Third derivative
|
gold
|
|
Fourth derivative
|
purple
|
It is
worthwhile to study these curves, checking that at any point
the red curve (first derivative) shows the slope of the blue curve,
and
crosses the x-axis wherever the blue curve
has a local max or min (note)
the green curve (second derivative) shows the
slope of the red curve
and whether the blue curve is concave up or down,
and so on….
In some cases I
give you formulas for the derivative, but in most of the examples I don’t. The point is for you to gain a solid intuitive understanding of
the relationship of the derivative to the function just from the graphs and knowing the conceptual definition of derivative.
Trivial
Examples
Constant function
There is no tangent line at (0,0) because the
graph has a corner there.
Polynomials
This graph also shows the tangent
line at , whose slope is
2 (note that ), and the tangent line at , whose slope is
1.
The (n-1)st
derivative of a polynomial of degree n
is always a straight line. The nth derivative is a horizontal line (y = 6 in this
case). Note that the
derivative is 0 at x = 0 but the
function does not have a local max or min there.
I scaled it by because it is hard to see the third derivative
for .
I fiddled with this
polynomial until I got the function and all four derivatives to be
separated from each other. All the roots
of the function and all its derivatives are real and all are shown. Isn’t this gorgeous?
Rational functions
Exponential functions
This function is its own derivative. This
means the slope of the tangent line to f
at the point is . The picture
shows
¨
the function f
¨
its tangent line at the point (0,1), which is a straight
line of slope 1
¨
its tangent line at the point (1,e), which is a straight
line of slope e, approximately 2.7.
This function fails to be its own first derivative,
but only by a little bit!
The famous “bell curve” is (below).
When you see it displayed the x and
y scales are almost always adjusted
so that the curve looks roughly like the one above for .
Trig functions
The derivative is just the sine curve moved to the left by units.
The curve is said to be phase-shifted.
The second derivative is the sine curve
upside down. If the sine curve
represents a sound, the negative sine curve cancels it. This is how sound-canceling earphones
work.
The third derivative is first derivative
upside down.
The sine
function is its own fourth derivative. In
other words, it is a solution (not the only one) of the differential equation .
I show this instead of tan x because it is prettier.
Log functions
In
abstractmath.org, “log(x)” means log
to the base e. Calculus books tend to call this “ln(x)”.
The second, third and fourth derivative of
this function are the same as the first, second and third derivatives of log(x).
Why?
Mixtures of
functions
The first derivative moves very slowly upward
from left to right. How can you tell
that from the formula?
. This
function is not defined as x = 1. Neither are any of its derivatives. Notice that when x gets big the fourth derivative is almost the same as the function.
The amplitude is a constant 1. Why? Because
the amplitude of sin is a constant 1 and f(x) is the “sin of something”. (That is conceptual thinking.)
The amplitude goes up like 2x.
The line y = 2x is plotted in black.
This
function is its own third derivative, so the blue curve is the derivative of
the green curve.
This function is a solution to the
differential equation . Compare the blue and purple curves.
