January 27, 2005 11:44
L24-ch04
Sheet number 1 Page number 127
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CHAPTER 4
Derivatives of Logarithmic, Exponential, and Inverse Trigonometric Functions EXERCISE SET 4.1 1. y = (2x − 5)1/3 ; dy/dx =
2 (2x − 5)−2/3 3
2. dy/dx =
−2/3 −2/3 2 1 2 + tan(x2 ) sec2 (x2 )(2x) = x sec2 (x2 ) 2 + tan(x2 ) 3 3
3. dy/dx =
2 3
4. dy/dx =
−1/2 −1/2 1 x2 + 1 d x2 + 1 −12x 6x 1 x2 + 1 √ = =− 2 x2 − 5 dx x2 − 5 2 x2 − 5 (x2 − 5)2 (x2 − 5)3/2 x2 + 1
x+1 x−2
3
5. dy/dx = x
√ 3 6. dy/dx = −
7. dy/dx =
2 − 3
−1/3
x − 2 − (x + 1) 2 =− (x − 2)2 (x + 1)1/3 (x − 2)5/3
(5x2 + 1)−5/3 (10x) + 3x2 (5x2 + 1)−2/3 =
1 2 x (5x2 + 1)−5/3 (25x2 + 9) 3
2 2x − 1 1 −4x + 3 + = 2 2 2/3 x x 3(2x − 1) 3x (2x − 1)2/3
15[sin(3/x)]3/2 cos(3/x) 5 [sin(3/x)]3/2 [cos(3/x)](−3/x2 ) = − 2 2x2
8. dy/dx = −
−3/2 −3/2 1 3 cos(x3 ) − sin(x3 ) (3x2 ) = x2 sin(x3 ) cos(x3 ) 2 2
6x2 − y − 1 dy dy − 6x2 = 0, = dx dx x 2 2 dy 2 + 2x3 − x = + 2x2 − 1, = − 2 + 4x (b) y = x x dx x 1 1 1 1 2 dy 2 2 = 6x − − y = 6x − − + 2x − 1 = 4x − 2 (c) From Part (a), dx x x x x x x
9. (a) 1 + y + x
dy 1 −1/2 dy √ y − cos x = 0 or = 2 y cos x 2 dx dx dy 2 (b) y = (2 + sin x) = 4 + 4 sin x + sin2 x so = 4 cos x + 2 sin x cos x dx dy √ = 2 y cos x = 2 cos x(2 + sin x) = 4 cos x + 2 sin x cos x (c) from Part (a), dx
10. (a)
11. 2x + 2y
dy dy x = 0 so =− dx dx y
12. 3x2 + 3y 2
dy dy dy 3y 2 − 3x y 2 − x2 = 3y 2 + 6xy , = 2 = 2 dx dx dx 3y − 6xy y − 2xy
dy dy + 2xy + 3x(3y 2 ) + 3y 3 − 1 = 0 dx dx dy dy 1 − 2xy − 3y 3 (x2 + 9xy 2 ) = 1 − 2xy − 3y 3 so = dx dx x2 + 9xy 2
13. x2
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