# Mathematics

34 6. DEFINITION OF THE DERIVATIVE

6.2. Exercises

(1) Suppose you know that the derivative of √ x is

1

2 √ x

for every x > 0. Then

lim x→9

√ x− 3 x− 9

= 1

a where a = .

(2) Suppose you know that the derivatives of x 1 3 is 13x

− 2 3 for every x 6= 0. Then

lim x→8

(x 8

) 1 3 − 1

x− 8 =

1

a where a = .

(3) Suppose you know that the derivative of ex is ex for every x. Then

lim x→2

ex − e2

x− 2 = .

(4) Suppose you know that the derivative of lnx is 1

x for every x > 0. Then

lim x→e

lnx3 − 3 x− e

= .

(5) Suppose you know that the derivative of tanx is sec2 x for every x. Then

lim x→π

4

tanx− 1 4x− π

= .

(6) Suppose you know that the derivative of arctanx is 1

1 + x2 for every x. Then

lim x→ √ 3

3 arctanx− π x− √

3 = .

(7) Suppose you know that the derivative of cosx is − sinx for every x. Then

lim x→π

3

2 cosx− 1 3x− π

= −1 a

where a = .

(8) Suppose you know that the derivative of cosx is − sinx for every x. Then

lim t→0

cos(π6 + t)− √ 3 2

t = −1

a where a = .

(9) Suppose you know that the derivative of sinx is cosx for every x. Then

lim x→−π/4

√ 2 sinx+ 1

4x+ π =

1

a where a = .

(10) Suppose you know that the derivative of sinx is cosx for every x. Then

lim x→ 7π

12

2 √

2 sinx− √

3− 1 12x− 7π

= 1− √ a

b where a = and b = .

(11) Let f(x) =

x2, for x ≤ 1 1, for 1 < x ≤ 3 5− 2x, for x > 3

. Then f ′(0) = , f ′(2) = , and f ′(6) = .

6.2. EXERCISES 35

(12) Suppose that the tangent line to the graph of a function f at x = 1 passes through the point (4, 9) and that f(1) = 3. Then f ′(1) = .

(13) Suppose that g is a differentiable function and that f(x) = g(x) + 5 for all x. If g′(1) = 3, then f ′(1) = .

(14) Suppose that g is a differentiable function and that f(x) = g(x+ 5) for all x. If g′(1) = 3, then f ′(a) = 3 where a = .

(15) Suppose that f is a differentiable function, that f ′(x) = −2 for all x, and that f(−3) = 11. Find an algebraic expression for f(x). Answer: f(x) = .

(16) Suppose that f is a differentiable function, that f ′(x) = 3 for all x, and that f(3) = 3. Find an algebraic expression for f(x). Answer: f(x) = .

36 6. DEFINITION OF THE DERIVATIVE

6.3. Problems

(1) Let f(x) = 1

x2 − 1 and a = −3. Show how to use the definition of derivative to find

Df(a).

(2) Let f(x) = 1√ x+ 7

. Show how to use the definition of derivative to find f ′(2).

(3) Let f(x) = 1√ x+ 3

. Show how to use the definition of derivative to find f ′(1).

(4) Let f(x) = √ x2 − 5. Show how to use the definition of derivative to findf ′(3).

(5) Let f(x) = √

8− x. Show how to use the definition of derivative to find f ′(−1). (6) Let f(x) =

√ x− 2. Show how to use the definition of derivative to find f ′(6).

(7) Let f(x) = x

x2 + 2 . Show how to use the definition of derivative to find Df(2).

(8) Let f(x) = (2×2 − 3)−1. Show how to use the definition of derivative to find Df(−2).

(9) Let f(x) = x + 2×2 sin 1

x for x 6= 0 and f(0) = 0. What is the derivative of f at 0 (if it

exists)? Is the function f ′ continuous at 0?

6.4. ANSWERS TO ODD-NUMBERED EXERCISES 37

6.4. Answers to Odd-Numbered Exercises

(1) 6

(3) e2

(5) 1

2

(7) √

3

(9) 4

(11) 0, 0, −2 (13) 3

(15) −2x+ 5

CHAPTER 7

TECHNIQUES OF DIFFERENTIATION

7.1. Background

Topics: rule for differentiating products, rule for differentiating quotients, chain rule, tangent lines, implicit differentiation.

7.1.1. Notation. We use f (n)(a) to denote the nth derivative of f at a.

7.1.2. Definition. A point a in the domain of a function f is a stationary point of f is f ′(a) = 0. It is a critical point of f if it is either a stationary point of f or if it is a point where the derivative of f does not exist.

Some authors use the terms stationary point and critical point interchangeably—especially in higher dimensions.

39

40 7. TECHNIQUES OF DIFFERENTIATION

7.2. Exercises

(1) If f(x) = 5x−1/2 + 6×3/2, then f ′(x) = axp + bxq where a = , p = , b = , and q = .

(2) If f(x) = 10 5 √ x3 +

12 6 √ x5

, then f ′(x) = axp + bxq where a = , p = ,

b = , and q = .

(3) If f(x) = 9×4/3 + 25×2/5, then f ′′(x) = axp + bxq where a = , p = , b = , and q = .

(4) If f(x) = 18 6 √ x+

8 4 √ x3

, then f ′′(x) = axp + bxq where a = , p = ,

b = , and q = .

(5) Find a point a such that the tangent line to the graph of the curve y = √ x at x = a has

y-intercept 3. Answer: a = .

(6) Let f(x) = ax2 + bx + c for all x. We know that f(2) = 26, f ′(2) = 23, and f ′′(2) = 14. Then f(1) = .

(7) Find a number k such that the line y = 6x + 4 is tangent to the parabola y = x2 + k. Answer: k = .

(8) The equation for the tangent line to the curve y = x3 which passes through the point (0, 2) is y = mx+ b where m = and b = .

(9) Let f(x) = 14x 4 + 13x

3− 3×2 + 74 . Find all points x0 such that the tangent line to the curve y = f(x) at the point (x0, f(x0)) is horizontal. Answer: x0 = , , and .

(10) In the land of Oz there is an enormous statue of the Good Witch Glinda. Its base is 20 feet high and, on a surveyor’s chart, covers the region determined by the inequalities

−1 ≤ y ≤ 24− x2 .

(The chart coordinates are measured in feet.) Dorothy is looking for her little dog Toto. She walks along the curved side of the base of the statue in the direction of increasing x and Toto is, for a change, sitting quietly. He is at the point on the positive x-axis 7 feet from the origin. How far from Toto is Dorothy when she is first able to see him? Answer: 5

√ a ft. where a = .

(11) Let f(x) = x− 32 x2 + 2

and g(x) = x2 + 1

x2 + 2 . At what values of x do the curves y = f(x) and

y = g(x) have parallel tangent lines? Answer: at x = and x = .

(12) The tangent line to the graph of a function f at the point x = 2 has x-intercept 10

3 and

y-intercept −10. Then f(2) = and f ′(2) = . (13) The tangent line to the graph of a function f at x = 2 passes through the points (0,−20)

and (5, 40). Then f(2) = and f ′(2) = .

(14) Suppose that the tangent line to the graph of a function f at x = 2 passes through the point (5, 19) and that f(2) = −2. Then f ′(2) = .

(15) Let f(x) =

x2, for x ≤ 1 1, for 1 < x ≤ 3 5− 2x, for x > 3

. Then f ′(0) = , f ′(2) = , and f ′(6) = .

(16) Suppose that g is a differentiable function and that f(x) = g(x+ 5) for all x. If g′(1) = 3, then f ′(a) = 3 where a = .

7.2. EXERCISES 41

(17) Let f(x) = |2− |x− 1|| − 1 for every real number x. Then f ′(−2) = , f ′(0) = , f ′(2) = , and f ′(4) = .

(18) Let f(x) = tan3 x. Then Df(π/3) = .

(19) Let f(x) = 1

x csc2

1

x . Then Df(6/π) =

π2

a

( π√ b − 1 )

where a = and b = .

(20) Let f(x) = sin2(3×5 + 7). Then f ′(x) = ax4 sin(3×5 + 7)f(x) where a = and f(x) = .

(21) Let f(x) = (x4 + 7×2 − 5) sin(x2 + 3). Then f ′(x) = f(x) cos(x2 + 3) + g(x) sin(x2 + 3) where f(x) = and g(x) = .

(22) Let j(x) = sin5(tan(x2+6x−5)1/2). ThenDj(x) = p(x) sinn(g(a(x)))f(g(a(x)))h(a(x)) ( a(x)

)r where

f(x) = ,

g(x) = ,

h(x) = ,

a(x) = ,

p(x) = ,

n = , and

r = .

(23) Let j(x) = sin4(tan(x3−3×2+6x−11)2/3). Then j ′(x) = 8p(x)f(g(a(x))) cos(g(a(x)))h(a(x)) ( a(x)

)r where

f(x) = ,

g(x) = ,

h(x) = ,

a(x) = ,

p(x) = ,

r = .

(24) Let j(x) = sin11(sin6(x3 − 7x+ 9)3). Then

Dj(x) = 198(3×2 + b) sinp(g(a(x)))h(g(a(x))) sinq(a(x))h(a(x)) ( a(x)

)r where

g(x) = ,

h(x) = ,

a(x) = ,

p = ,

q = ,

r = , and

b = ,

(25) Let f(x) = (x2 + sinπx)100. Then f ′(1) = .

(26) Let f(x) = (x2 − 15)9(x2 − 17)10. Then the equation of the tangent line to the curve y = f(x) at the point on the curve whose x-coordinate is 4 is y = ax+b where a = and b = .

42 7. TECHNIQUES OF DIFFERENTIATION

(27) Let f(x) = (x2−3)10(x3+9)20. Then the equation of the tangent line to the curve y = f(x) at the point on the curve whose x-coordinate is −2 is y = ax + b where a = and b = .

(28) Let f(x) = (x3−9)8(x3−7)10. Then the equation of the tangent line to the curve y = f(x) at the point on the curve whose x-coordinate is 2 is y = ax + b where a = and b = .

(29) Let f(x) = (x2 − 10)10(x2 − 8)12. Then the equation of the tangent line to the curve y = f(x) at the point on the curve whose x-coordinate is 3 is y = ax+b where a = and b = .

(30) Let h = g ◦ f and j = f ◦ g where f and g are differentiable functions on R. Fill in the missing entries in the table below.

x f(x) f ′(x) g(x) g′(x) h(x) h′(x) j(x) j ′(x)

0 −3 1 1 −32 1 0 32 0

1 2

(31) Let f = g ◦ h and j = g · h where g and h are differentiable functions on R. Fill in the missing entries in the table below.

x g(x) g′(x) h(x) h′(x) f(x) f ′(x) j(x) j ′(x)

0 2 −4 −6 3

1 −2 4 −4 2

2 4 4 13 24 4 19

Also, g(4) = and g′(4) = .

(32) Let h = g ◦ f , j = g · f , and k = g + f where f and g are differentiable functions on R. Fill in the missing entries in the table below.

x f(x) f ′(x) g(x) g′(x) h(x) h′(x) j(x) j ′(x) k(x) k′(x)

−1 −2 4 4 −2

0 0 0 −1 1

1 2 2 0 6

(33) Let y = log3(x 2 + 1)1/3. Then

dy

dx =

2x

a(x2 + 1) where a = .

(34) Let f(x) = ln (6 + sin2 x)10

(7 + sinx)3 . Then Df(π/6) =

a

5 where a = .

(35) Let f(x) = ln(lnx). What is the domain of f? Answer: ( , ). What is the equation of the tangent line to the curve y = f(x) at the point on the curve whose

x-coordinate is e2? Answer: y − a = 1 b

(x− e2) where a = and b = .

(36) Find when y = (tanx)sinx for 0 < x < π/2. Then dy

dx = (tanx)sinx(f(x) + cosx ln tanx)

where f(x) = .

7.2. EXERCISES 43

(37) Find when y = (sinx)tanx for 0 < x < π/2. Then dy

dx = (sinx)tanx(a+ f(x) sec2 x) where

a = and f(x) = .

(38) d

dx

√ x lnx = xp(1 + g(x)) where p = and g(x) = .

(39) If f(x) = x3ex, then f ′′′(x) = (ax3 + bx2 + cx + d)ex where a = , b = , c = , and d = .

(40) Let f(x) = x2 cosx. Then (ax2 + bx+ c) sinx+ (Ax2 +Bx+C) cosx is an antiderivative of f(x) if a = , b = , c = , A = , B = , and C = .

(41) Let f(x) = (x4 − x3 + x2 − x + 1)(3×3 − 2×2 + x − 1). Use the rule for differentiating products to find f ′(1). Answer: .

(42) Let f(x) = x3/2 − x 3x− x1/2

. Then f ′(4) = 9

a where a = .

(43) Find a point on the curve y = x2

x3 − 2 where the tangent line is parallel to the line 4x +

6y − 5 = 0. Answer: ( , ). (44) Let f(x) = 5x cosx − x2 sinx. Then (ax2 + bx + c) sinx + (Ax2 + Bx + C) cosx is an

antiderivative of f(x) if a = , b = , c = , A = , B = , and C = .

(45) Let f(x) = (2x−3) cscx+(2+ 3x−x2) cotx cscx. Then ax 2 + bx+ c

sinx is an antiderivative

of f(x) if a = , b = , and c = .

(46) Let f(x) = x2 − 10 x2 − 8

. Find the equation of the tangent line to the curve y = f(x) at

the point on the curve whose x-coordinate is 3. Answer: y = ax + b where a = and b = .

(47) Let f(x) = (x4 + x3 + x2 + x + 1)(x5 + x3 + x + 2). Find the equation of the tangent line to the curve y = f(x) at the point on the curve whose x-coordinate is −1. Answer: y = ax+ b where a = and b = .

(48) Let y = x2 − 2x+ 1 x3 + 1

. Then dy

dx

∣∣∣∣ x=2

= 2

a where a = .

(49) Let f(x) = x− 32 x2 + 2

and g(x) = x2 + 1

x2 + 2 . At what values of x do the curves y = f(x) and

y = g(x) have parallel tangent lines? Answer: at x = and .

(50) Let f(x) = x sinx. Find constants a, b, A, and B so that (ax+ b) cosx+ (Ax+B) sinx is an antiderivative of f(x). Answer: a = , b = , A = , and B = .

(51) Find d

dx

( 1

x

d2

dx2

( 1

1 + x

)) = a

bx+ 1

x2(1 + x)b where a = and b = .