Thomas Calculus 13th [Solutions]

202 Chapter 3 Derivatives
43.
1 4x 1 4x dy 1 4x 4x 1 4x 4x 1 4x 1 4x 4x
y xe e [ x(4 e ) e (1)] (4 e ) xe e e xe
4 16 dx 4 16 4 4
44.
1 1 1 1
2 2/ x 2 2x dy 2 2 2x 2x 2x 2/ x
y x e x e x [(2 x ) e ] e (2 x) (2 2 x) e 23 (1 x)
dx
45. ln(sin 2 dy 2(sin )(cos )
y
2cos
d
2
sin sin
2cot
46. ln(sec 2
dy 2(sec )(sec tan )
y
d
2
sec
2 tan
47.
y
2
ln
x
2 2
log x
dy 1 x 2
2 2 ln 2 dx ln 2 x
2 (ln 2)
2
x
48.
y
ln(3x
7) dy
log 1 3 3
5(3x
7)
ln 5 dx ln 5 3x 7 (ln 5)(3x
7)
t dy t t
49. y 8 8 (ln 8)( 1) 8 (ln 8) 50.
dt
y
2t dy 2t 2t
9 9 (ln 9)(2) 9 (2ln 9)
dt
51.
3.6 dy
2.6 2.6
y 5x 5(3.6) x 18x
dx
52.
53.
y 2x 2 dy
2 1 2 1
2 2 x 2x
dx
y ( x 2 2
2) ln ln( x
2) ( 2) ln( y
2) ( 2) 1
y
x 2
(1) ln( 2)
dy x 2
( x 2) [ln( x
dx
2) 1]
54.
y /2 /2
2(ln x x
) ln y ln[2(ln x x
) ] ln(2) x y
ln(ln ) 0 x (ln(ln )) 1
2 x y 2 ln x
x 2
1
x
1 1 x/2 x/2
y ln(ln x) 2(ln x) (ln x) ln(ln x)
1
2ln x 2 ln x
55.
1 2 1 2 1/2
y sin 1 u sin (1 u )
1
2
2 1/2
dy (1 u ) ( 2 u)
u u u 1
du
2 1/2
1 (1 u ) 1 u 1 (1 u ) u 1 u u 1 u 1 u
2 2 2 2 2 2
,
0 < u < 1
56.
y
sin
1
2
3/2
1 1 1 1/2 dy v
1 1 v
sin v
1
3/2
v
dv
1 ( v ) 2v 1 v
2v
v vv1
2 1
1/2 2 3/2 1 3/2 v 1
2v
v
57.
1
1 x
2
1 2 1
1
y ln(cos x)
y
1
cos x 1 x cos x
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