Thomas Calculus 13th [Solutions]

Section 3.8 Derivatives of Inverse Functions and Logarithms 171
4. (a)
(c)
2 2
y 2x x 1 y
2
df
dx x
1
1
x y f ( x) x
2
2
1
df
dx
5
x
50
4x
20,
x 5
1 1/2
x
1
2 2 20
x 50
(b)
5. (a)
3
3
f ( g( x)) x x,
3 3
g( f ( x))
x x
(b)
(c)
2
f ( x) 3 x f (1) 3, f ( 1) 3;
1 2/3
g ( x) x g (1) 1 , g ( 1) 1
3 3 3
3
(d) The line y = 0 is tangent to f ( x)
x at
(0, 0); the line x = 0 is tangent to
(0, 0).
g( x)
3
x at
6. (a)
(c)
1 1/3 3
h( k( x)) ((4 x) ) x,
4
3
k( h( x)) 4 x 1/3
x
4
2
h ( x) 3x
h (2) 3, h ( 2) 3;
4
4 2/3
k ( x) (4 x) k (2) 1 , k ( 2) 1
3 3 3
(d) The line y = 0 is tangent to h( x ) x at
4
1/3
(0, 0); the line x = 0 is tangent to k( x) (4 x)
at (0, 0).
3
(b)
7.
1
df 2 df
1 1
3x 6x 8.
dx dx df
x f (3) 9
dx x 3
1
df
df
2x
4 1 1
dx dx df 6
x f (5) dx x 5
9.
df
dx
1 1
df
1 1
dx
df
1
x 4 x f (2) dx x 2 3
3
10.
dg
dx
1 1
dg
1 1
dx
dg 2
x 0 x f (0) dx x 0
11. y ln 3x x y 1 (3) 1 1 1
3x
x
12. y
ln3x
1 y 1 1 1
2 3
(ln3 x) 3x
x(ln3 x)
2
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