102 3. LIMITS, DERIVATIVES, RULES, AND THE MEANING OF THE DERIVATIVE
-=2
=2
=2
6-6
-1 1
Figure 3.8: e inverse secant (light) and inverse cosecant (dark) functions.
Knowledge Box 3.11
Inverse trigonometric derivatives
f .x/ f
0
.x/
sin
1
.x/
1
p
1 x
2
cos
1
.x/
1
p
1 x
2
tan
1
.x/
1
1 C x
2
cot
1
.x/
1
1 C x
2
sec
1
.x/
1
jxj
p
x
2
1
csc
1
.x/
1
jxj
p
x
2
1
3.3. DERIVATIVES OF THE LIBRARY OF FUNCTIONS 103
PROBLEMS
Problem 3.43 Find the derivative of each of the following functions.
1. f .x/ D
4
p
x
2. g.x/ D
x
3:1
3. h.x/ D
x
4. r.x/ D sin.x/ C cos.x/
5. s.x/ D 4e
x
6. q.x/ D tan
1
.x/ C =2
Problem 3.44 Find the tangent line to each of the following functions at the indicated point.
1. f .x/ D
p
x at x D 1
2. g.x/ D sin.x/ at x D =3
3. h.x/ D cos.x/ at x D =4
4. r.x/ D tan
1
.x/ at x D 0
5. s.x/ D sin
1
.x/ at x D 0:5
6. q.x/ D ln.x/ at x D ln.2/
Problem 3.45 Find an inverse function for each of the following functions.
1. f .x/ D x
2
C 2x C 1
2. g.x/ D 13x 27
3. h.x/ D tan.3x C 2/
4. r.x/ D
2x C 1
x 1
5. s.x/ D e
2x
6. q.x/ D
1
x
Problem 3.46 Find an inverse of the function f .x/ D x
2
on the interval .1; 0 .
Problem 3.47 Which of the following functions have universal inverses? Justify your answer.
1. f .x/ D ln.x/
2. g.x/ D e
x
3. h.x/ D 2x C 5
4. r.x/ D x
2
5. s.x/ D x
3
6. q.x/ D .2x C 5/
3
Problem 3.48 Find the universal inverse of
f .x/ D ax C b
when a and b are real numbers.
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