11 ALGEBRA

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Университет имени Сулеймана Демиреля

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Check Yourself 39

Find the derivative of each function.

1. f(x) = x2 ln x 2. g(x)= ln 7 x5

4.f(x) = ln(x2 – 5x + 1) 5. h(x) = [ln(x2 + x + 1)]2

7.f(x) = log2(x2 + 5x – 1) 8. g(x) = log3[ln(x2 + 1)]

Answers
1. x(2ln x + 1)				2.		5
1. x(2ln x + 1)				2.		7x
						7x
4.			2x – 5	5. 2ln(x2 + x + 1) (				2x+1	)
4.	x	2	– 5x+1	5. 2ln(x2 + x + 1) (				2	)
	x		– 5x+1				x + x+1
7.			2x+5	8.			2x
7.	(x2 +5x – 1)ln2			8.	(x2 +1)[ln (x2 +1)]ln3

3.h(x) = (ex + ln x)2

6.f(x) = ln(ex + 2)

9.g(x) = log2(ex + x – 1)

3. 2(ex + ln x) (ex + x1)

6. ex ex +2

9. 2(ex +1)log(ex + x – 1) (ex + x – 1)ln10

3. Logarithmic Differentiation

Sometimes the task of finding the derivative of a complicated function involving products, quotients, or powers can be made easier by first applying the laws of logarithms to simplify it. This technique is called logarithmic differentiation. Let us look at some examples.

EXAMPLE 121 f(x) = x(3x – 1)(x2 + 3) is given. Find the first derivative of f by using logarithmic differentiation.

Solution f(x) = x(3x – 1)(x2 + 3)

ln f(x) = ln [x(3x – 1)(x2 + 3)]

ln f(x) = In(x) + ln(3x – 1) + ln(x2 + 3)

ln (ab) = ln a + ln b

f (x) = 1 +

+ 2x

a, b > 0

f (x) x 3x – 1 x2 +3

f (x)= f (x)(

1 +

)

3x – 1

x2 +3

f (x)= x(3x – 1)(x2 +3)(

)

3x – 1

x2 +3

(take the logarithms is of both sides) (write logarithms as sums)

(take derivatives of both sides)

(isolate f (x))

(substitute for f(x))

Exponential and Logarithmic Functions

229

EXAMPLE 122 Find the derivative of the function f (x)= x5 3 x2 +1. (2x+1)7

Solution

ln( ab) = ln a – ln b

ln ap = p ln a a, b > 0

First find the logarithms of both sides of the expression:

x2 +1

3 ln

1 ln(x2

ln f (x)= ln x5

+1) 7 ln(2 x+1).

(2x+1)

Now differentiate both sides of the equation with respect to x:

f (x)

1 + 1

7 2

x2 +1

(2x+1)

f (x)

x2 +1

x5 3

f (x)= f(x)

2x+1

3x +3

2x+1

3x +3

(2x+1)

If we had not used logarithmic differentiation here, finding the derivative would have been a long and complicated process.

EXAMPLE 123 Given that x > 0, find the derivative of f(x) = xx.

Solution First find the logarithms of both sides of the expression: ln f(x) = ln xx = x ln x.

Differentiate both sides of the equation with respect to x:

f (x) = x ln x+ x (ln x) =1+ ln x f (x)

f (x)= f(x)(1+ ln x)= xx(1+ ln x).

Check Yourself 40

Find the derivative of each function by using logarithmic differentiation.

f(x) = (3x – 1)5(x3 + 1)6

2. f (x)= ex2 +1 (x2 – 1)10

f (x)=

4 x2 +1

– 1

g(x)= x

h(x) = xln x

f (x)=(ln x)

ln x

Answers

18x2

20x

ex2 +1

(x2 – 1)10

(3x – 1)

+1)

2x+

–

3x –

x +1

x2 +1

1– ln x

2 ln x

ln x

1– ln(ln x)

ln x

(ln x)

–

xln

– 1

– 1)(x +1)

230	Algebra 11

B. INTEGRALS OF EXPONENTIAL FUNCTIONS

BASIC INTEGRATION FORMULAS - 2

		a.		1 dx = ln\| x\|+ c
				x
		b.		u'(x) dx	= ln\| u(x)\|+ c
		b.		u(x)	= ln\| u(x)\|+ c

	124 1y		dy =?
EXAMPLE			dy =?

Solution 1			dy = ln\| y\|+ c
		y

EXAMPLE 125 2xx2 dx =?

Solution	Using formula 2b:	2x	dx = ln	x2	+ c = ln x2 + c	since ( x2 ) = 2x

		2
		x

EXAMPLE 126 cossin xx dx =?

Solution

cos x

dx = ln|sin x| + c

since (sin x) = cos x

sin x

127

dx =?

EXAMPLE

Solution

dx = 4 ln|x – 3| + c

since (x – 3) = 1

x 3

Check Yourself 41

Evaluate the integrals.

5x5

+2x2 +3x – 5

–

Answers

5x4

a. ln| x|+ x +

+ c

4 +2x+3ln| x|+ x

+ c

5 ln|5x+1|+c

2x2

Exponential and Logarithmic Functions

231

BASIC INTEGRATION FORMULAS - 3

		a.	ex dx = ex + c
		b.	ax dx =			ax	+c
		b.	ax dx =			ln a	+c
						ln a

	128 3x dx =?
EXAMPLE

		x		3x
Solution 3			dx =		+ c
Solution 3			dx =	ln3	+ c
				ln3

129 4 5x

dx =?

EXAMPLE

4 5x

Solution

4 5x

dx = 4 5x

dx =

+ c

ln5

130

4ex

dx =?

EXAMPLE

Solution

4ex

dx = 4ex c

EXAMPLE 131

74x 3

dx =?

4 x

)

4x 3

Solution

74x 3

dx =

(7 3)

dx =

+ c =

+ c

4 ln7

ln7

132

5ex 1

dx =?

EXAMPLE

Solution

5ex 1

dx = 5e ex dx = 5e ex + c = 5ex 1 + c

Properties of Exponents:

am an = am+n

am = am – n

(am)n = amn

232	Algebra 11

	133 e3x 1	dx =?
EXAMPLE		dx =?
			3	x			3x 1
Solution e3x 1		dx = e (e3 )x dx = e	(e	)		+c = e		+ c.
Solution e3x 1		dx = e (e3 )x dx = e		3	)	+c = e	3	+ c.
			ln(e		)		3

Note

We can make the following generalizations of formulas 3a and 3b:

EXAMPLE 134 e4x 1dx =?

Solution By (1) above, e4x 1 dx = 41 e4x 1 +c.

EXAMPLE 135 53x 1 dx =?

Solution By (2) above,

53x 1 dx=

1 53x 1

+ c=

53x 1

+ c.

3 ln5

3ln5

Check Yourself 42

Evaluate the integrals.

a. 5 3x dx

b. (2x 3x ) dx

c. 65x 1 dx

x 3

d. 3 5x 3 + 4

e. 3e3x dx

2e(2x 3) dx

g. 24x 4 dx

h. 4ex

Answers

5 3x

65x 1

3 5x 3

4x 3

ln3 + c

–

+ c

ln2

ln3

5ln6

ln5

3ln4

e. e

2x-3

24x 6

22ex 2 1

+ c

ln2 + c

e ln2 + c

Exponential and Logarithmic Functions

233

EXERCISES 3.4

A.Derivatives of Exponential and Logarithmic Functions

1.Differentiate the functions.

a. f(x) = 3ex			b. f(x) = e3x – 1
c. f(x) = e	x2	– 1	–2x
c. f(x) = e			d. f(x) = e
e. f(x) = 2x			f. f (x)=( 1)x
			3

g.f(x) = ex 3x

i. f(x) = x2 + 2ex

k. f (x)= ex – 1

	m. f (x)=	1
	m. f (x)=	ex +1
		ex +1

h. f(x) = xex

ex +1 j. f (x)= ex – 1

l. f(x) = (ex + x)100

n. f(x) = (ex + x)(2ex – 1)

	f (x)=	ex + e–x						1
o.		ex + e–x				p. f (x)= e2x
o.			3			p. f (x)= e2x
			3
								x –1

q. f (x)= e x+1						r. f (x)= e x+1
s.		–2t	2		x2–1	t. f (x)=	6x – 1
s.	f(x) = e		+ x	e		t. f (x)=	3x +1
u. f(x) = 3x2 + 4x						v. f (x)=		53x+1
u. f(x) = 3x2 + 4x						v. f (x)=		2 x
							x + e
w. f (x)= 2 x2 –x–1						x. f(x) = 2ex + 2 3x

2. Differentiate the functions.
a. f(x) = 3ln x	b. f(x) = ln 4x
c. f(x) = 3ln 4x	d. f(x) = 3ln (2x + 1)
e. f(x) = ln x7	f.	f(x) = ln ñx
g. f(x) = log3 x	h.	f(x) = log1/2 x
i. f(x) = xlog x	j.	f(x) = log (x2	+ 1)
		2

k. f(x) = ln (4x2 – 6x + 3)


l. f (x)= ln x+1				m. f (x)= ln	x – 1
					x+1
		x – 1
n. f(x) = x2ln x				o. f (x)= ln x2
p. f (x)= ln x+ x				q. f(x) = ln(ñx – 1)–2
r. f(x) = ln(x2 – x)				s. f(x) = ex ln x
t.	f (x)=	ln(x2 x)		u. ln (x+1)(x – 2)

			ex	x – 1
v.	f (x)= log		x	w. f (x)= log3		x +1
			x+1			x – 1

x.f (x)=(ex – log 2 x2 )3

y.f (x)= x2 – log3 ex

z.f (x)=(log(1+ ex ))3

3.Find the derivative of each function by using logarithmic differentiation.

a.f(x) = (2x – 1)7(x4 – 3)11

b.f(x) = x(x + 1)(x2 + 1)

c.f (x)= 3 x2 ex2 –1 (x3 – x)–2

	d. f (x)=	4+3x2
	d. f (x)=	3 x2 +1
		3 x2 +1

e. f(x) = xñx

f. f(x) = (ln x)x+1

234	Algebra 11

4.Find the equation of the tangent line to the graph of y = ex2–1 at the point P(1, 1).

5.Find the equation of the tangent line to the curve y = ex + e–x at the point (0, 2).

6.Find the equation of the tangent line to the graph of y = x2ln x at the point (1, 0).

Mixed Problems

7.Evaluate the integrals, using the basic formulas for integration.

a. xx3		dx	2
a. xx3		dx	b. 3xx3 dx
	3

c. ( x2 + x +1+ x) dx d. (sin x+cos x) dx

x+1

+3x

4x+1

8.Evaluate the integrals, using the basic formulas for integration.

a.	e2x dx	b.	e5x dx
c.	3e2x dx	d.	5e7x 2 dx
e.	7ex 2 dx	f.	22x+1 dx
g.	5x dx	h.	62x 1 dx
i.	43x 4 dx	j.	33x dx
k.	10x 1 dx	l.	4 32x–1 dx

9.Evaluate the integrals using the substitution method.

a.				1		dx	b. cos		dx
		x			3
								x
c.			x		dx		d.	5	dx
								3x+1
			2
			x
e.	esin x cos x dx

10.Evaluate the integrals using the substitution method.

a.	ln x		dx		b. cot x dx

		x
c.		ex e x			d.	ex
				dx			dx
						ex 3
		ex +e x

Exponential and Logarithmic Functions

235

11. Evaluate the integrals by using partial fractions.

x4 +2x2 + x

2x+1

7x 6

x2 + x 1

3x2 4x+1

4x2 +5x+4

3x+1

x2 +1

(x+1)(x+2)

x 1

2x 1

x2 1 dx

x2 2x 3

x2 +3x 1

(x 1)3

x2 +8x+15

1 x

(1+ x)

11x2 +5x+12

(x 1)(x2 + x+2)

m.

x2 +2x+2

x3 1

n.

3x4 – 16x3 +19x2

5x – 4

x3 4x2 +1

o.

2x+3

(x+1)(x2 +2)

p. dx 2 1 4x+ x

12. Evaluate the integral of each radical function.

a.	1	dx	b.	9x2 +1 dx
	2
	1+ x
c.	x2 9 dx		d.	dx
				x2 9

13.Evaluate the integral of each function by using the substitution t = tan 2x .

	sin x				1
a.			dx	b.				dx
						tan x+sin x
	3+cos x
c.	1	dx		d.		3 sin x	dx
	sin x
					1 cos x

236	Algebra 11

CHAPTER REVIEW TEST 3A

1. An exponential function is of the form f(x) = ax.

6. Which statement is false?

calculate f(–2).

A) 4–3 + 4–3 + 4–3 + 4–3 = 4–2

Given f (3)= 8

B) 4–6 = 2–12

A) 1

C) 2

D) 4

E) 8

C) (4–2)3 = 4–5

D) 4–1 + 4–1 = 2–1

E) 2

4–3 = 2–5

2. Simplify

( a)

( a

)

( a)

( a) .

A) a5

B) –a5

C) a25

D) –a25

E) a15

7. Solve 841 = x

2125 for x.

A) 1

D) 2

E) 4

3. ax = 2 is given. What is

a2x – a–2x

8. Which of the following is not an exponential

A) 1

B) 3

E) 15

function for x ?

A) f(x) = 22x

B) f(x) = 3–2x

C) f(x) = –7x/3

D) f(x) = (–2)x

E) f(x) = 11–x/4

4. Which graph is shown in	y
the figure opposite?		25
		1	x	9. Calculate	29 57	.
	0.04		x	9. Calculate	106	.
	0.04		2		106
-2	0		2
				A) 400	B) 40		C) 4	D) 0.4	E) 0.04

A) y = 5x		B) y = 2x		C) y = 2–x
D) y = 4–x			E) y = 5–x
					10. Which of the	following is	equivalent to
					5+2 6 ?
5. f(x) = 8	7x is given. Calculate f –1(8).				A) 7 + ñ6	B) ñ3 + ñ2	C) 1 + ñ6
A) 56	B) 8	C) 7	D) 1	E) 0	D) ñ3 + 1	E) ñ2 + 2

Chapter Review Test 3A

237

11. Which of the following is greater than 1?

		1		–	–2			10		10
A)	(	)2		B) (0.3)			C) (	3	)	3
		D) (	1 4			1	0.6
			4)		E) (		)
			4)		E) (	6	)

12. Calculate	4 –	12	.
12. Calculate			.
	7 – 4 3
A) 2	B) 5		C) 7	D) 4	E) 6

13. Calculate		3103 – 3102		.
13. Calculate				.
			952
A)	1	B)	1		C)	2	D)	2	E) 3
	3		9			3		9

14. What is half of 420?
A) 220	B) 410	C) 210	D) 239	E) 419

15. Evaluate	51.1	+	90.4	.
	125–0.3		9–0.1
A) 25.01	B) 23		C) 25.9 D) 28 E) 34

16.	3 22n–1	– 6 22 n–3	= 2n	is given. Find n.
	3 22n+1
	A) 1	B) –2		C) 2	D) 3	E) –3

17. How	many digits	are there	in the	number
510 44 63?
A) 10	B) 11	C) 12	D) 13	E) 14

18. Evaluate	( 2 – 5)2 + (–5)2 +	2.
A) 2ñ2	B) 0 C) 2ñ2 – 10	D) 10 E) –10

19.Which statement is not necessarily true for the exponential function f(x) = ax?

A) f(x) is injective	B) f(x) is bijective
C) f(x) is increasing	D) f(x) is surjective

E) f(x) is positive

20. Given that	a = 6	4	, which of the following is an
20. Given that	a = 6	2	, which of the following is an
		2
integer?
A) a6	B) a2		C) a3	D) a4	E) a5

238	Algebra 11

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