7 inverse functions. 7.4 derivatives of logarithmic functions in this section, we find: the...

7INVERSE FUNCTIONSINVERSE FUNCTIONS

7.4Derivatives of

Logarithmic Functions

In this section, we find:

The derivatives of the logarithmic functions y = logax

and the exponential functions y = ax.

INVERSE FUNCTIONS

We start with the natural logarithmic

function y = ln x.

We know it is differentiable because it is the inverse of the differentiable function y = ex.

DERIVATIVES OF LOG FUNCTIONS

1(ln )

dx

dx x

Formula 1

DERIVATIVES OF LOG FUNCTIONS Formula 1—Proof

Let y = ln x.

Then, ey = x.

Differentiating this equation implicitly with respect to x, we get:

So,

1y dyedx

1 1y

dy

dx e x

Differentiate y = ln(x3 + 1).

To use the Chain Rule, we let u = x3 + 1. Then, y = ln u. So,

23

2

3

1 1(3 )

1

3

1

dy dy du dux

dx du dx u dx x

x

x

DERIVATIVES OF LOG FUNCTIONS Example 1

In general, if we combine Formula 1

with the Chain Rule, as in Example 1,

we get:

1 '( )(ln ) or ln ( )

( )

d du d g xu g x

dx u dx dx g x

DERIVATIVES OF LOG FUNCTIONS Formula 2

Find .

Using Formula 2, we have:

ln(sin )d

xdx

1ln(sin ) (sin )

sin1

cos cotsin

d dx x

dx x dx

x xx

DERIVATIVES OF LOG FUNCTIONS Example 2

Differentiate .

This time, the logarithm is the inner function.

So, the Chain Rule gives:

( ) lnf x x

1 212

1 1'( ) (ln ) (ln )

2 ln1

2 ln

df x x x

dx xx

x x

DERIVATIVES OF LOG FUNCTIONS Example 3

Find .

1 212

12

1 1 1ln

12 22

2 1 ( 1) ( 2)2

1 22 ( 1) 5

( 1)( 2) 2( 1)( 2)

d x d xxdx dxx xx

x x xx

x xx x x

x x x x

1ln

2

d x

dx x

DERIVATIVES OF LOG FUNCTIONS E. g. 4—Solution 1

If we first simplify the given function using the

laws of logarithms, the differentiation becomes

easier:

This answer can be left as written. However, if we used a common denominator,

it would give the same answer as in Solution 1.

12

1ln ln( 1) ln( 2)

2

1 1 1

1 2 2

d x dx x

dx dxx

x x

DERIVATIVES OF LOG FUNCTIONS E. g. 4—Solution 2

Find the absolute minimum value of:

f(x) = x2 ln x

The domain is (0, ∞) and the Product Rule gives:

Therefore, f ’(x) = 0 when 2 ln x = -1, that is, ln x = -½, or x = e-½.

Example 5DERIVATIVES OF LOG FUNCTIONS

2 1'( ) 2 ln (1 2ln )f x x x x x x

x

Also, f ’(x) > 0 when x > e-½

and f ’(x) < 0 when 0 < x < e-½.

So, by the First Derivative Test for Absolute Extreme Values,

is the absolute minimum.

Example 5

(1/ ) 1/(2 )f e e

DERIVATIVES OF LOG FUNCTIONS

Discuss the curve

y = ln(4 – x2)

using the guidelines

of Section 4.5

DERIVATIVES OF LOG FUNCTIONS Example 6

A. The domain is:

DERIVATIVES OF LOG FUNCTIONS Example 6

2 2{ 4 0} { 4}

{ | | 2}

( 2,2)

x x x x

x x

B. The y-intercept is f(0) = ln 4.

To find the x-intercept we set:

y = ln(4 – x2) = 0

We know that ln 1 = loge1 = 0 (since e0 = 1).

So, we have 4 – x2 = 1 x2 = 3. Therefore, the x-intercepts are .

DERIVATIVES OF LOG FUNCTIONS Example 6

3

Since f(-x) = f(x), f is even

and the curve is symmetric

about the y-axis.

DERIVATIVES OF LOG FUNCTIONS Example 6

D. We look for vertical asymptotes at

the endpoints of the domain.

Since 4 – x2 → 0+ as x → 2- and also as x → -2+, we have:

by Equation 8 in Section 7.3. So, the lines x = 2 and x = -2 are vertical

asymptotes.

DERIVATIVES OF LOG FUNCTIONS Example 6

2 2

2 2lim ln(4 ) lim ln(4 )x x

x x

and

DERIVATIVES OF LOG FUNCTIONS Example 6

E.

f ’(x) > 0 when -2 < x < 0 and f ’(x) < 0 when 0 < x < 2.

Thus, f is increasing on (-2, 0) and decreasing on (0, 2).

2

2'( )

4

xf x

x

DERIVATIVES OF LOG FUNCTIONS Example 6

F. The only critical number is x = 0.

f ’ changes from positive to negative at 0.

Therefore, f(0) = ln 4 is a local maximum by the First Derivative Test.

G.

f ’’(x) < 0 for all x. So, the curve is concave downward on (-2, 2)

and has no inflection point.

DERIVATIVES OF LOG FUNCTIONS Example 62

2 2

2

2 2

(4 )( 2) 2 ( 2 )''( )

(4 )

8 2

(4 )

x x xf x

x

x

x

H. Using that information, we sketch

the curve.

DERIVATIVES OF LOG FUNCTIONS Example 6

© Thomson Higher Education

Find f ’(x) if f(x) = ln |x|.

Since

it follows that:

Thus, f ’(x) = 1/x for all x ≠ 0.

ln if 0( )

ln( ) if 0

x xf x

x x

1if 0

( )1 1

( 1) if 0

xxf x

xx x

DERIVATIVES OF LOG FUNCTIONS Example 7

The result of Example 7 is worth

remembering:

1ln

dx

dx x

DERIVATIVES OF LOG FUNCTIONS Equation 3

The corresponding integration formula

is:

Notice that this fills the gap in the rule for integrating power functions:

The missing case (n = -1) is supplied by Formula 4.

DERIVATIVES OF LOG FUNCTIONS Formula 4

1ln | |dx x C

x

1

11

nn xx dx C n

n

if

Find, correct to three decimal places,

the area of the region under the hyperbola

xy = 1 from x = 1 to x = 2.

DERIVATIVES OF LOG FUNCTIONS Example 8

© Thomson Higher Education

?

Using Formula 4 (without the absolute value

sign, since x > 0), we see the area is:

DERIVATIVES OF LOG FUNCTIONS Example 8

2

2

1 1

1ln

ln 2 ln1

ln 2

0.693

A dx xx

© Thomson Higher Education

Evaluate:

We make the substitution u = x2 + 1 because the differential du = 2x dx occurs (except for the constant factor 2).

Example 9DERIVATIVES OF LOG FUNCTIONS

2 1

xdx

x

Thus, x dx = ½ du and

Notice that we removed the absolute value signs because x2 + 1 > 0 for all x.

DERIVATIVES OF LOG FUNCTIONS Example 9

1 12 22

212

212

ln | |1

ln | 1|

ln ( 1)

x dudx u C

x u

x C

x C

We could use the properties of

logarithms to write the answer as:

However, this isn’t necessary.

2ln 1x C

DERIVATIVES OF LOG FUNCTIONS Example 9

Calculate:

We let u = ln x because its differential du = dx/x occurs in the integral.

When x = 1, u = ln 1 = 0; when x = e, u = ln e = 1. Thus,

Example 10DERIVATIVES OF LOG FUNCTIONS

1

lne xdx

x

121

1 00

ln 1

2 2

e x udx u du

x

The function f(x) = (ln x)/x in Example 10

is positive for x >1.

Thus, the integral represents the area of the shaded region in the figure.

DERIVATIVES OF LOG FUNCTIONS

Calculate:

First, we write tangent in terms of sine and cosine:

This suggests that we should substitute u = cos x since, then, du = -sin x dx and so sin x dx = -du.

DERIVATIVES OF LOG FUNCTIONS Example 11

tan x dx

sintan

cos

xx dx dx

x

Thus,

DERIVATIVES OF LOG FUNCTIONS Example 11

sintan

cosln | |

ln | cos |

x duxdx dx

x uu C

x C

Since

the result of Example 11 can also be

written as:

DERIVATIVES OF LOG FUNCTIONS

ln | cos | ln(1/ | cos |) ln | sec |x x x

tan ln | sec |x dx x C

Formula 5

Formula 7 in Section 7.3 expresses

a logarithmic function with base a in terms

of the natural logarithmic function:

GENERAL LOG FUNCTIONS

lnlog

lna

xx

a

Since ln a is a constant, we can differentiate

as follows:

ln 1 1(log ) (ln )

ln ln lna

d d x dx x

dx dx a a dx x a

GENERAL LOG FUNCTIONS

1(log )

lna

dx

dx x a

Formula 6

Using Formula 6 and the Chain Rule,

we get:

GENERAL LOG FUNCTIONS Example 12

10log (2 sin )

1(2 sin )

(2 sin ) ln10

cos

(2 sin ) ln10

dx

dxd

xx dx

x

x

From Formula 6, we see one of the main

reasons that natural logarithms (logarithms

with base e) are used in calculus:

The differentiation formula is simplest when a = e because ln e = 1.

GENERAL LOG FUNCTIONS

In Section 7.2, we showed that the derivative

of the general exponential function

f(x) = ax, a > 0

is a constant multiple of itself:

EXPONENTIAL FUNCTIONS WITH BASE a

0

1'( ) '(0) '(0) lim

hx

h

af x f a f

h

where

We are now in a position to show that

the value of the constant is f ’(0) = ln a.

EXP. FUNCTIONS WITH BASE a Formula 7

( ) lnx xda a a

dx

We use the fact that e ln a = a:

EXP. FUNCTIONS WITH BASE a

ln (ln )

ln

( ) ( ) (ln )

( ) (ln )

ln

x a x a x

a x

x

d d da e e a x

dx dx dx

e a

a a

Formula 7—Proof

In Example 6 in Section 3.7, we considered

a population of bacteria cells that doubles

every hour.

We saw that the population after t hours is:

n = n02t

where n0 is the initial population.

EXP. FUNCTIONS WITH BASE a

Formula 7 enables us to find

the growth rate:

EXP. FUNCTIONS WITH BASE a

0 2 ln 2tdnn

dt

Combining Formula 7 with the Chain

Rule, we have:

EXP. FUNCTIONS WITH BASE a Example 13

2 2 2

2

(10 ) 10 (ln10) ( )

(2 ln10) 10

d dx x x

dx dx

x x

The integration formula that follows

from Formula 7 is:

EXP. FUNCTIONS WITH BASE a Example 13

1ln

xx aa dx C a

a

EXP. FUNCTIONS WITH BASE a Example 14

55

00

5 0

22

ln 2

2 2

ln 2 ln 231

ln 2

xxdx

The calculation of derivatives of

complicated functions involving products,

quotients, or powers can often be simplified

by taking logarithms.

The method used in the following example is called logarithmic differentiation.

LOGARITHMIC DIFFERENTIATION

Differentiate:

We take logarithms of both sides and use the Laws of Logarithms to simplify:

3 / 4 2

5

1

(3 2)

x xy

x

23 14 2ln ln ln( 1) 5ln(3 2)y x x x

LOGARITHMIC DIFFERENTIATION Example 15

Differentiating implicitly with respect to x gives:

Solving for dy/dx, we get:

2

1 3 1 1 2 35

4 2 3 21

dy x

y dx x xx

2

3 15

4 1 3 2

dy xy

dx x x x

LOGARITHMIC DIFFERENTIATION Example 15

Since we have an explicit expression for y, we can substitute and write:

3 / 4 2

5 2

1 3 15

4 3 2(3 2) 1

dy x x x

dx x xx x

LOGARITHMIC DIFFERENTIATION Example 15

If we hadn’t used logarithmic differentiation

in Example 15, we would have had to use

both the Quotient Rule and the Product Rule.

The resulting calculation would have been horrendous.

LOGARITHMIC DIFFERENTIATION Note

1. Take natural logarithms of both sides

of an equation y = f(x) and use the Laws

of Logarithms to simplify.

2. Differentiate implicitly with respect to x.

3. Solve the resulting equation for y’.

LOGARITHMIC DIFFERENTIATION—STEPS

If f(x) < 0 for some values of x, then ln f(x)

is not defined.

However, we can write | y | = | f(x) | and

use Equation 3.

We illustrate this procedure by proving the general version of the Power Rule—as promised in Section 3.3.

LOGARITHMIC DIFFERENTIATION

If n is any real number and f(x) = xn,

then

Let y = xn and use logarithmic differentiation:

Thus,

Hence,

1'( ) nf x nx

THE POWER RULE

ln ln ln 0n

y x n x x 'y n

y x

1'n

ny xy n n nx

x x

Proof

You should distinguish carefully

between:

The Power Rule [(d/dx) xn = nxn-1], where the base is variable and the exponent is constant.

The rule for differentiating exponential functions [(d/dx) ax = ax ln a], where the base is constant and the exponent is variable

LOGARITHMIC DIFFERENTIATION Note

In general, there are four cases for

exponents and bases:

1

( ) ( )

( )

1. ( ) 0 and areconstants

2. ( ) ( ) '( )

3. (ln ) '( )

4. To find ( / [ ( )] , logarithmic

differentiation can be used, as in the next example.

b

b b

g x g x

g x

da a b

dxdf x b f x f x

dxda a a g x

dx

d dx f x

LOGARITHMIC DIFFERENTIATION

Differentiate .

Using logarithmic differentiation, we have:

ln ln ln

' 1 1(ln )

2

1 ln 2 ln'

2 2

x

x

y x x x

yx x

y x x

x xy y x

x x x

LOGARITHMIC DIFFERENTIATION E. g. 16—Solution 1

xy x

Another method is to write:

ln ln( ) ( ) ( ln )

2 ln

2

x x x x x

x

d d dx e e x x

dx dx dx

xx

x

LOGARITHMIC DIFFERENTIATION E. g. 16—Solution 2

ln( )x x xx e

We have shown that, if f(x) = ln x,

then f ’(x) = 1/x.

Thus, f ’(1) = 1.

Now, we use this fact to express the number e as a limit.

THE NUMBER e AS A LIMIT

From the definition of a derivative as a limit,

we have:

0 0

0

0

1

0

(1 ) (1) (1 ) (1)'(1) lim lim

ln(1 ) ln1lim

1lim ln(1 )

lim ln(1 )

h x

x

x

x

x

f h f f x ff

h xx

x

xx

x

THE NUMBER e AS A LIMIT

As f ’(1) = 1, we have:

Then, by Theorem 8 in Section 2.5 and the continuity of the exponential function, we have:

1

0lim ln(1 ) 1x

xx

1/1

0lim ln(1 )1 ln(1 ) 1

0 0lim lim(1 )

xx

xx x x

x xe e e e x

THE NUMBER e AS A LIMIT

1

0lim(1 ) x

xe x

Formula 8

Formula 8 is illustrated by:

The graph of the function y = (1 + x)1/x.

A table of values for small values of x.

THE NUMBER e AS A LIMIT

If we put n = 1/x in Formula 5, then

n → ∞ as x → 0+.

So, an alternative expression for e is:

1lim 1

n

ne

n

THE NUMBER e AS A LIMIT Formula 9