Math 1210-001
Friday Jan 29
WEB L112
Finish 2.2, derivatives; begin 2.3, shortcut rules for taking derivatives.
Summary of this week so far:
1) For a function f defined on the closed interval a, b , the average rate of change of f on a, b is
defined by the difference quotient
f b Kf a
.
bKa
output units
ft
people coulombs
The units for the average rate of change are
, e.g.
,
,
, etc.
input units
sec hour
sec
Geometrically, the average rate of change is the slope of the ("secant") line through the points
a, f a , b, f b on the graph of f.
2) For a function f x defined on an open interval containing c, the derivative of f at c is denoted by f# c
("f prime at c"). It is defined as the limit of the average rates of change of f on intervals having c has either
the left or right endpoint, as the length of those intervals goes to zero. In other words,
f# c d lim
h/0
f cCh Kf c
.
h
In terms of rates of change, f# c is called the instantaneous rate of change of f ; the units of f# c are
output units
, just as for the average rates of change. Geometrically, f# c represents the slope of the
input units
tangent line to the graph of f at the point c, f c on the graph, obtained as the limit of the secant line
slopes.
3) We say that f is differentiable at c if f# c exists. Otherwise we say that f is not differentiable at c.
Finish Wednesday's notes, beginning with Exercise 3. Then continue with today's notes.
Derivative shortcuts (Section 2.3)
A helpful notation for derivative is the letter D with a subscript indicating the letter we're using for the
variable in the function we're differentiating. In other words
Dt f t = f# t
Dxg x = g# x
etc. In the discussion of Wednesday's notes we have shown:
Dx 1 = 0
Dx x = 1
Dx x2 = 2 x
Dx f x C g x = Dx f x C Dx g x
Dx k f x = k Dx f x if k is a constant.
Exercise 1) Use the limit definition of derivative to show Dx x3 = 3 x2 .
Exercise 2) Use the limit definition of derivative to show that Dx xn = n x n K 1 for every counting
number n = 1, 2, 3,...
Exercise 3) Use the rules we know so far, to compute
Dx 5 x10 C p x4 C 3 x3 K 27 x2 C 17
Rules for taking derivatives of products and quotients!
Alas, unlike the limit theorems for products and quotients of functions - where the limit of a product (or
quotient) is the product (or quotient) of the limits, it is almost never true that the derivative of a product of
two functions is the product of the derivatives; nor is the derivative of a quotient the quotient of the
derivatives.
Exercise 4) Is Dx x$x = Dx x $Dx x ?
Exercise 5) Is Dx
x
x
=
Dx x
Dx x
?
Theorem (We'll understand why these rules are true next week, probably on Monday):
Product rule: Dx f x g x
Quotient rule: Dx
f x
g x
= f# x g x C f x g# x
=
f# x g x K f x g# x
g x
2
Exercise 6) Compute
Dx x2 C 1 2 x C 3
two ways: Once with the product rule, and once by first expanding the product function into a cubic.
Verify that your answers agree.