Chapter 3

48 | T r a n s c e n d e n t a l F u n c t i o n s

Chapter 3
Transcendental Functions
3.1 The Natural Logarithm Function
The power of calculus, both that of derivatives and integrals, has already been
amply demonstrated. Yet we have only scratched the surface of potential
applications. To dig deeper, we need to expnnd the class of functions with which we
can work. That is the object of this chapter.
We begin by asking you to notice a peculiar gap in our knowledge of
derivatives.

Is there a function whose derivative is

In other words, is there an antiderivative

The First Fundamental Theorem of Calculus states that the accumulation
function

is a function whose derivative is f(x), provided that f is continuous on an interval I
that contains a and x. In this sense, we can find an antiderivative of ally continuous
function. The existence of an antiderivative does not mean that the antiderivative can
be expressed in terms of functions that we have studied so far. In this chapter we
will introduce and study a number of new functions.

Definition 3.1 Natural Logarithm Function
The natural logarithm function, denoted by ln, is defined by

The domain of the natural logarithm function is the set of positive real numbers.


The diagrams in Figure 1 indicate the geometric meaning of in x, The natural
logarithm or natural log) function measures the area under the curve y = 1/t between
1 and x if x > 1 and the negative of the area if 0 < x < 1. The natural logarithm is an
accumulation function because it accumulates area under the curve

.

Figure 1
. Clearly, In x is well defined for

is not defined for

because this

definite integral does not exist over an interval that includes 0.
And what is the derivative of this new function? Just exactly what we want.

Derivative of the Natural Logarithm Function
From the First Fundamental Theorem of CaLculus, we have

This can be combined with the Chain Rule. If
differentiable, then

EXAMPLE 1 Find
SOLUTION Let

Then

and if

f

is


EXAMPLE 2 Find
SOLUTION

This problem makes sense, provided that

.

which s positive provided that
Thus, the domain of

EXAMPLE 3

is

Now

or

. On this domain,

Show that

SOLUTION
wo cases are to be considered.

and

, and so

We know that for every differentiation formula there is a corresponding integration
formula. Thus. Example 3 implies that

or with replacing x,

This fills the long-standing gap in the Power Rule:
which we had to exclude the exponent

EXAMPLE 4

Find

SOLUTION

Let

EXAMPLE 5

Evaluate

so

. Then

, from

.


SOLUTION

Let

so

. Then

Thus, by the Second Fundamental Theorem of Calculus,

For the above calculation to be valid,

must never be 0 on the interval

It is easy to see that this is true.
When the integrand is the quotient of two polynomials (that is, a rational
function) and the numerator is of equal or greater degree than the denominator,
always divide the denominator into the numerator first,

EXAMPLE 6

Find

SOLUTION
By long division

Hence,

Properties of the Natural Logarithm
The next theorem lists several important properties of the natural log function.


Theorem 3.1
If

and

are positive numbers and r is any rational number, then

(i)

(ii)

(ii)

(iv)

Proof
(i)
(ii)

Since, for

and

it follows from the theorem about two functions with the same derivative that

To determine C let

Finally, let
(iii)

, obtaining In

.

Replace a by

in (ii) to obtain

Thus,

Applying (ii) again, we get

(iv)

Since, for

and

We get

Finally, let

. Thus,


Logarithmic Differentiation
The labor of differentiating expressions involving quotients, products, or powers can
often be substantially reduced by first applying the. natural logarithm function and
using its properties. This method, called logarithmic differentiation, is illustrated in
Example 8.
EXAMPLE 8

Differentiate

SOLUTION
First we take natural logarithms; then we differentiate implicitly with respect to x.

Thus,

Example 8 could have been clone directly, without first taking logarithms, and we
suggest you try it. You should be able to make the two answers agree.

Trigonometric Integrals
Some trigonometric integrals can be evaluated using the natural log function.

EXAMPLE 9

Evaluate

SOLUTION
Since

we can make the substitution

obtain

Similarly,

EXAMPLE 10

Evaluate

.

,

, to


SOLUTION

For this one we use the trig identity

Then

Exercises 3.1
In problems 1-4, find the indicated derivative
1.
2.
3.
4.
In problems 5-10 , find the integrals
5.
6.
7.
8.
9.
10.

.


3.2 Inverse Functions and Their Derivatives
In this section, we study the general problem of reversing (or inverting) a
function. Here is the idea.
A function f takes a number x from its domain D and assigns to it a single
value y from its range R. If we are lucky, as in the case of the two functions graphed
in Figures 2 and 3, we can reverse f; that is, for any given y in R, we can
unambiguously g back and find the x from which it came, This new function that
takes y and assigns x to it is denoted by

. Note that its domain is R and its range

is Di It is called the inverse off or simply f-inverse. Here we are using the superscript
in a new way. The symbol

oes not denote.

. , as you might have expected

We, and all mathematicians, use it to name the inverse function.

Figure 3

Figure 2

Sometimes, we can give a formula for
(see

Figure

2).

Similarly,

If
if

then
then

(Figure 3). In each caset we simply solve the equation that
determines f for x in terms of y. The result is

.

But life is more complicated than these two examples indicate. Not every
function can be reversed in an unambiguous way. Consider

for

example. For a given y there are two is that correspond to it (Figure 4). The function


is even worse, For each y there are infinitely many x's that
correspond to it (Figure 5), Such functions do not have inverses; at least, they do not
unless we somehow restrict the set of x-values, a subject we will take up later.

Figure 4

Figure 5

Existence of Inverse Functions
It would be nice to have a simple criterion for deciding whether a function f has an
inverse. One such criterion is that the function be one-to-one; that is,

implies

This is equivalent to the geometric condition that every horizontal
line meet the graph of y =

in at most one point. But, in a given situation, this

criterion, may be very hard to apply, since it demands that we have complete
knowledge of the graph. A more practical criterion that covers most examples that
arise in this book is that a function be strictly monotonic. By this we mean that it is
either increasing or decreasing on its domain.

Theorem 3.2
If

is strictly monotonic on its domain, then

Proof Let

and

monotonic,
implies

has an inverse.

be distinct numbers in the domain off, with
or

. Since

. Either way,

is

Thus,

which means that f is one-to-one and therefore has

an inverse.
This is a practical result, because we have an easy way of deciding whether a
differentiable function f is strictly monotonic. We simply examine the sign of


EXAMPLE 11 Show that

has an inverse.

SOLUTION
for all . Thus, f is increasing on the whole real line and so it
has an inverse there.
We do not claim that we can always give a formula for

. In the example

just considered, this would require that we be able. to solve

for

. There is a way of salvaging the notion of inverse for functions that do not have
inverses on their natural domain. We simply restrict the domain to a set on which
the graph is either increasing or decreasing. Thus, for
restrict the domain to

(

, we may

would also work). For

restrict the domain to the interval

, we

. Then both functions have inverses (see

Figure 5), and we can even give a formula for the first one :

Figure 6
If
and

has an inverse

then

also has an inverse, namely, . Thus, we may call

a pair of inverse functions. One function undoes (or reverses) what the other

did that is,

EXAMPLE 2
Show that

has an inverse, find a formula for

, and verify the

results in the box above,
SOLUTION
Since f is an increasing function, it has an inverse. To find
for , which gives

and

, we solve

Finally, note that


The Graph of
Suppose that f has an inverse. Then

Consequently,

and

determine the same (x, y) pairs and so

have identical graphs. However, it is conventional to use x as the domain vari able
for functions, so we now inquire about the graph of

(note that we have

in the roles of x and y). A little thought convinces us that to inter- change the roles of
x and v on a graph is to reflect the graph across the line
Thus the graph of
the line

.

is just the reflection of the graph of

across

(Figure 6).

Figure 6
A related matter is that of finding a formula for

To do it, we first find

and then replace y by x in the resulting formula. Thus, we propose the following
three-step process for finding
Step 1 : Solve the equation
Step 2 : Use

for

in terms of .

to name the resulting expression in .

Step 3 : Replace y by x to get the formula fo
Before trying the three-step process on a particular function f, you might think we
should first verify that f has an inverse. However, if we can actually carry out the


first step and get a single x for each y, then
this for

we get

does exist. (Note that when we try
, which immediately shows that

does not exist, unless, of course, we have restricted the domain to eliminate one of
the two signs,

or .)

EXAMPLE 3
Find a formula dor

if

SOLUTION
Here are the three steps for this example.
Step 1:

Step 2:

Step 3:

Derivatives of Inverse Functions
We conclude this section by investigating the relationship between the derivative of a
function and the derivative of its inverse. Consider first what happens to a line
when it is reflected across the line y = x. As the left half of Figure 7 makes clear,
reflected into a line
provided

; moreover, their respective slopes
. If

and

is

are related by

happens to be the tangent line to the graph of f at


the point (c, d), then

is the tangent line to the graph of

at the point (d, c) (see

the right half of Figure 7), We are led to the conclusion that

Figure 7
Pictures are sometimes deceptive, so we claim only to have made the following
result plausible, For a formal proof, see any advanced calculus book.

Theorem 3.3 Inverse Function Theorem
Let

be differentiable and strictly monotonic on an interval I, If

certain x in , then
range of

differentiable at the corresponding point

and

The conclusion to Theorem B is often written symbolically as

at a
in the


EXAMPLE 4

Let

as in Example 1. Find

SOLUTION
Even though we cannot find a forma for
corresponds to

in this case, we note that

, and,since

,

Exercises 3.2
In Problems 1 – 5, find a formula for
dan

and then verify that

.
1.
2.
3.
4.
5.
6. If
and

then f has an inverse (why?) Let

)

Find

(a)
(b)
(c)

3.3 The Natural Exponential Function
The graph of

was obtained at the end of Section 3.1 and

is reproduced in Figure 1.The natural logarithm function is differentiable (hence
continuous) and increasing on its domain

): its range is

. It

is, in fact, precisely the kind of function studied in Section 3.2, and therefore has an
inverse In-1 with domain

and range

that it is given a special name and a special symbol

. This function is so important


Figure 1
Definition 3. 2
The inverse of In is called the natural exponential function and is denoted by exp,
Thus,

It follows immediately from this definition that
1.
2.

for all

Since exp and In are inverse functions, the graph of
= In x reflected across the line

(Figure2)

But why the name exponentiaifunction? You will see.

Figure 2

is just the graph of y


Definition 3.3
The letter

denotes the unique positive real number such that In

.

Figure 3 illustrates this definition: the area under the graph of
and

is 1. Since In e = 1, it is also true that

between
. The number ,

like , is irrational. Its decimal expansion is known to thousands of places; the first
few digits are

Figure 3
Now we make a crucial observation, one that depends only on facts already
demonstrated: (1) above and Theorem 3.1 (i). If

is any rational number,

Let us emphasize the result. For rational r, exp r is identical with

What was

introduced in the most abstract way as the inverse of the natural logarithm, which
itself was defined by an integraL has turned out to be a simple power.
But what if r is irrational'? Here we remind you of a gap in all elementary
algebra books. Never are irrational powers defined in anything approaching a
rigorous manner. What is meant by

? You will have a hard time pinning that

number down., based on elementary algebra, But we must pin it down if we are to
talk of such things as
for all

. Guided by what we learned above, we simply define

(rational or irrational) by

Note that (1) and (2) at the beginning of this section now take the following form:


Theorem 3.4
Let

and

any real numbers. Then

and

Proof To prove the first, we write

=

The second fact is proved similarly,

The Derivative of
Since exp and ln are inverses, we know from Theorem 3.2 (ii) that
differentiable. To find a formula for

is

we could use that the Alternatively, let

so that

Now differentiate both sides with respect to . Using the Chain Rule, we obtain

Thus,

We have proved the remarkable fact that

Thus,

is a solution of the differential equation

differentiable, then the Chain Rule yields

EXAMPLE 5

is its own derivative: that is,

Find

If

is


SOLUTION

Using

we obtain

EXAMPLE 6 Find
SOLUTION

EXAMPLE 7
Let

. Find where

is increasing and decreasing and where it is concave

upward and downward. Also, identify all extreme values and points of inflection.
Then, sketch the graph of

.

SOLUTION

and

Keeping in mind that

for all x, we see that

and

. Thus,

for all

is decreasing on

and has its minimum value at
Also
of

for
is

for

of
and

and concave upward on
Since

and increasing on

for

so the graph

and has a point of inflection at
the line

asymptote. This information supports the graph in Figure 4.

is a horizontal


Figure 4
The derivative formula
, or, with

EXAMPLE 8

Let

EXAMPLE 9

Evaluate

SOLUTION

Let

replacing ,

Evaluate

SOLUTION

automatically yields the integral formula

EXAMPLE 10

Evaluate

so

so

Then

. Then


SOLUTION

Let

so

. Then

Thus, by the Second Fundamental Theorem of Calculus,

The last recall can be obtained directly with a calculator.

EXAMPLE 11

Evaluate

SOLUTION

Think of

Although the symbol

Let

, so

will largely supplant exp

Then

throughout the rest of this book,

exp occurs frequently in scientific writing, especially when the exponent

is

complicated. For example, in statistics, one often encounters the normal probability
density function, which is


Exercises 3.3
In Problem 1 – 5, find
1.
2.
3.
4.
5.

(Hint: Use implicit differentiation)

In Problem 6 – 10, find each integral
6.
7.
8.
9.
10.


3.4 General Exponential and Logarithmic Functions
We defined

, and and all other irrational powers of e in the previous section.

But what about

and similar irrational powers of other numbers? In fact,

we want to give meaning to

for

a rational number, then

and

any real number. Now, if

is

. But we also know that

This suggests the definition of the exponential function to the base .

Definition 3.4
For

and any real number ,

Of course, this definition will he appropriate only if the usual properties of exponents
are valid for it, a matter we take up shortly. To shore up our confidence in the
definition, we use it to calculate

(with a little help from our calculator):

Your calculator may give a result that differs slightly from 9. Calculators use
approximations for

and

, and they round to a fixed number of decimal places

(usually about 8)

Properties of
Theorem 3.5 Properties of Exponents
If

, and

and y are real numbers, then

Proof We will prove (ii) and (iii), leaving the others for you.


Theorem 3.5 Exponential Function Rules

Proof

The integral formula follows immediately from the derivative formula.

EXAMPLE 12
SOLUTION

EXAMPLE 13
SOLUTION

Find
We use the Chain Rule with

Find

if


EXAMPLE 14
SOLUTION

Find
Let

so

Then

The Function
Finally, we are ready to make a connection with the
algebra, We note that if

then

rithms that you studied in
is a decreasing function; if

, it is an increasing function, as you may check by considering the derivative.
In either case,

has an inverse. We call this inverse the logarithmic function to the

base . This is equivalent to the following definition.

Definition 3.6
Let

be a positive. number different from 1. Then

Historically, the most commonly used base was base 10, and the resulting logarithms
were called common logarithms. But in calculus and all of advanced mathematics,
the significant base is . Notice that

, being the inverse of

, is just

another synthol for In; that is,

We have come full circle (Figure 1), The. function

, which we introduced in

Section 6.1, has turned out to he an ordinary logarithm, but t a rather special base,
Now observe that if
, so that

, then

from which we conclude that

.


Also,

EXAMPLE 15
SOLUTION

If

find

Let

The Functions

.

and apply the Chain Rule.

and

We have just learned that

What about

? For

rational, we proved the Power Rule in Chapter 2, which

says that
Now ‘ve assert that this is true even if a is irrational. To see this, write

The corresponding rule for integrals also holds even if

Finally, we consider
for

is irrational.

a variable to a variable power. There is a formula

, but we do not recommend that you memorize it. Rather, we. suggest that

you learn two methods for finding it, as illustrated below.

EXAMPLE 16

If

find

by two different methods


SOLUTION
Method 1 We may write

Thus using the Chain Rule and the Product Rule,

Method 2 Recall the logarithmic differentiation technique from Section 3.1,

EXAMPLE 17

If

find

SOLUTION

EXAMPLE 18

If

find

SOLUTION We use logarithmic differentiation.

.


Exercises 3.4
In Problem 1 – 10, find the indicated derivative or integral.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.

Chapter 3

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