114 3. LIMITS, DERIVATIVES, RULES, AND THE MEANING OF THE DERIVATIVE
Problem 3.72 Find the derivatives of the following functions with the most appropriate tech-
nique(s).
1. f .x/ D
sin.x/
cos.x/
5
2. g.x/ D x ln.cos
2
.x/ C 1/
3. h.x/ D
x
2
C 1
5
x
7
C 1
4. q.x/ D x
2
ln.x
2
C 1/
5. r.x/ D x 5
x
2
C1
6. s.x/ D sin.x/ ln.x/
x
2
C 1
7. a.x/ D
x
3
C 1
121
x
x C 4
8. b.x/ D ln
x
3
C 1
x
2
C 2
Problem 3.73 Find the tangent lines to
f .x/ D
1
x
2
C 1
where x D ˙1. Give your answer in point-slope form.
Problem 3.74 Find the tangent line to
g.x/ D x e
x
2
where x D 0. Give your answer in point-slope form.
Problem 3.75 Find the tangent line to
h.x/ D tan
1
.x=2/
where x D
p
3. Give your answer in point-slope form.
Problem 3.76 Remembering that
tan.x/ D
sin.x/
cos.x/
use the quotient rule to verify the rule for the derivative of tan.x/.
3.4. THE PRODUCT, QUOTIENT, RECIPROCAL, AND CHAIN RULES 115
Problem 3.77 Remembering that
cot.x/ D
cos.x/
sin.x/
use the quotient rule to verify the rule for the derivative of cot.x/.
Problem 3.78 Remembering that
sec.x/ D
1
cos.x/
use the reciprocal rule to verify the rule for the derivative of sec.x/.
Problem 3.79 Remembering that
csc.x/ D
1
sin.x/
use the reciprocal rule to verify the rule for the derivative of csc.x/.
Problem 3.80 Derive the reciprocal rule by applying the quotient rule to
g.x/ D
1
f .x/
Problem 3.81 By using the product rule a couple times and simplifying, verify the triple prod-
uct rule:
.
f .x/g.x/h.x/
/
0
D f .x/g.x/h
0
.x/
C f .x/g
0
.x/h.x/
C f
0
.x/g.x/h.x/
Problem 3.82 Find the derivative of
f .x/ D x
5
.x C 1/
4
.x C 2/
3
116 3. LIMITS, DERIVATIVES, RULES, AND THE MEANING OF THE DERIVATIVE
Problem 3.83 Would taking the logarithm of both sides first make Problem 3.82 any easier?
Problem 3.84 Find the derivative of
g.x/ D x
2
sin.x/ e
x
Problem 3.85 Find the derivative of
h.x/ D
p
x C 1 tan.x/ ln.x/
Problem 3.86 By using the chain rule as many times as needed, verify the triple chain rule:
.
f .g.h.x///
/
0
D
f
0
.g.h.x/// g
0
.h.x// h
0
.x/
Problem 3.87 Find the derivative of
f .x/ D
.
sin.ln.x//
/
5
Problem 3.88 Find the derivative of
g.x/ D
.
ln.e
x
C 1/
/
3
Problem 3.89 Find the derivative of
g.x/ D ln
sin
4
x
4
C 1
Problem 3.90 Prove that
.
f .ax C b/
/
0
D a f
0
.ax C b/
for any function f .x/ with a derivative.
Problem 3.91 Find the derivative of f .x/ D sin.x/ cos.x/ with the chain rule instread of the
product rule. Hint: use a trig identity.
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