Thomas Calculus 13th [Solutions]

Section 3.8 Derivatives of Inverse Functions and Logarithms 177
log
85.
2 t (ln t)/(ln 2) dy (ln t)/(ln 2) 1 1 log
3 3 3 (ln 3) (log
2 t
y
ln 2 2 3)3
dt t t
86.
y
3ln
ln t
3ln(log 2 t) ln 2 dy 3 3
3log 1 1 1
8(log 2 t)
ln8 ln 8 dt ln 8 (ln t)/(ln 2) t ln 2 t(ln t)(ln8) t(ln t)(ln 2)
87.
ln 2
ln 2 ln8 ln( t ) 3ln 2 (ln 2)(ln t)
dy
y log 1
2(8 t ) 3 ln t
ln 2 ln 2
dt t
88.
ln 3 sin
ln ( ) t
sin t
ln(3 ) (sin )(ln 3)
(sin t)(ln 3)
t e t t t dy
y t log3 e t sin t sin t t cos t
ln 3 ln3 ln 3
dt
x x y
89. ( 1) ln ln( 1) ln( 1) 1
x
y x y x x x x ln( x 1) y ( x 1) x ln( x 1)
y ( x 1) x 1
90.
( x 1) ( x 1) y
1 1 ( x 1)
y x ln y ln x ( x 1)ln x ( x 1) ln x ln x 1 y x 1 1 ln x
y x x x
91.
t
1/2 t t /1 t/2 1 dy 1 1 ln 1 dy t
y t ( t ) t ln y ln t t ln t (ln t)
t t t ln t 1
2 y dt 2 2 t 2 2 dt
2 2
92.
1/2 1/2
y t t t ( t ) ( ) 1/2 1 1 1/2 1/2
ln y ln t t
( t )(ln t dy
) (ln ) 1 ln t 2 dy tln t 2
y dt 2 t t t t 2 t dt
t 2 t
93. y (sin x x
) ln y ln(sin x x
) x ln(sin x y
) cos x
x
ln(sin ) (sin ) [ln(sin ) cot ]
y
x sin x
x y x x x x
94.
sin x
sin x
y 1
sin x x(ln x)(cos x)
y x ln y ln x (sin x)(ln x) (sin x) (cos x)(ln x)
y x x
sin x sin x x(ln x)(cos x)
y x
x
95.
ln x 2 y
1
ln x
y x , x 0 ln y (ln x) 2(ln x) y ( x ) ln x
y x x
2
96.
ln x
y
1 1 ln(ln x)
y (ln x) ln y (ln x) ln(ln x) (ln x) d (ln x) ln(ln x)
1
y ln x dx x x x
ln x ln(ln x) 1
y (ln x)
x
97. ( g f )( x) x g( f ( x)) x g ( f ( x)) f ( x) 1
98.
lim 1 x
n
lim 1
n
n
n
1
( n/ x)
( n/ x)
x
x
e for any x > 0.
n
99. The derivative of x at x 0 is given by
n
(0 h) n 1
lim lim h . For n 2, n 1 1, so
h 0
h
h 0
n 1
lim h 0.
h 0
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