6
Friday, September 2
Derivatives
Definition (Rates of Change). The average rate of change of a function f is defined to be the slope of
secant lines, or lines that meet a graph in at least two points.
Rateavg
=
=
=
slope of secant line
change in f
change in x
f (c + h) − f (c)
h
The expression
f (c + h) − f (c)
h
is called a difference quotient. Average rate works well, but data is lost in the ”smoothing” effect.
It is more natural to look at tangent lines, or lines that meet a graph at a point, but do not cross the
graph. Thus, we take a limit as h → 0:
f 0 (c)
=
=
slope of tangent line at x = c
f (c + h) − f (c)
lim
h→0
h
The limiting value is called the derivative of the function f at the point x = c. The slope of the tangent
line is often called the instantaneous rate of change.
Note (Notation of derivatives). There are many different ways to denote the derivative:
dy dy df
0
0
y Dx [y]
.
f (x)
dx dx
dx x=c
If y = x2 + x, then
dy
d
=
x2 + x = 2x + 1
dx
dx
dy = 7.
dx x=3
Example (Finding derivatives and tangent lines). What is the equation for the line tangent to the graph
of f at the specified point?
(1) f (x) = 6 at c = −1.
(2) f (x) = 3x2 + x − 1 at c = 1.
(3) f (x) =
(4) f (x) =
3
at c = −2.
x
√
2x + 1 at c = 4.
1
(5) f (x) = √ at c = 3.
x
(6) f (x) =
1
at c = 2.
x2
Example. Find
dy .
dx x=x0
(1) y = 1 − 3x at x0 = 2.
(2) y =
x−1
at x0 = 1.
x+1
Example 6.1 (Finding average rates of change).
(1) Find the slope of the secant line that meets f (x) = x2 − 1 at x = 1 and x = 1.1. Find the slope of the
tangent line at x = 1.
(2) Find the average rate of change of f (x) = x(2 + 3x) from x = 0 to x = 1/3. Find the instantaneous rate
of change at x = 0.
Definition (Alternate definition of the derivative). The derivative can also be expressed as the limit
f 0 (c) = lim
x→c
f (x) − f (c)
x−c
Example. Find the derivatives of the functions using the alternate definition.
(1) f (x) = 5x − 3
(2) f (x) =
(3) f (x) =
1
x−1
√
x−1
Note. The derivative can possibly not exist. There are two primary cases for when this happens.
(1) Corners
(2) Vertical tangents
f (x) = |x|
f (x) =
√
3
x