Exercises-III
Exercise 1
Calculate the derivatives of the following functions:
1. y =
tg x – 1
sec x
: Ans. y0 = sin x + cos x
2. y = log r1 + sin 1 – sin x x: Ans. y0 = cos 1 x
3. y = log tg π4 + x2 : Ans. y0 = cos 1 x 4. f(x) = sin(log x): Ans. f0(x) = cos(log x x)
5. f(x) = tg(log x): Ans. f0(x) = sec2(log x)
x
6. y = log 1 + x
1 – x
. Ans. y0 = 2
1 – x2
7. y = log 1 + x2
1 – x2 : Ans. y0 = 1 -4xx4 8. y = log x2 + x : Ans. y0 = 2 xx2 ++ 1 x
9. y = log x3 – 2x + 5 : Ans. y0 = 3×2 – 2
x3 – 2x + 5
10. y = x log x: Ans. y0 = log x + 1
11. y = log(x + p1 + x2): Ans. y0 = p1 + 1 x2
12. y = log(log x): Ans. y0 = 1
x log x
13. f(x) = log r1 + 1 – x x. Ans. f0(x) = 1 -1×2
14. f(x) = log
px2 + 1 – x
px2 + 1 + x. Ans. f0(x) = -p1 + 2 x2
15. y = pa2 + x2 – a log a + pa2 + x2
x
. Ans. y0 =
pa2 + x2
x
16. y = log(x – px2 + a2) –
px2 + a2
x
. Ans. y0 =
px2 + a2
x2
17. y = –
cos x
2 sin2 x +
1 2
log tg x
2
: Ans. y0 = 1
sin3 x 18. y =
1 2
tg2 x+ log cos x: Ans. y0 = tg3 x
19. y = e4x+5. Ans. y0 = 4e4x+5 20. y = 7×2+2x: Ans. y0 = 2(x + 1)7×2+2x log
21. y = aepx. Ans. y0 = a
2pxe
px 22. y = ex 1 – x2 : Ans. y0 = ex 1 – 2x – x2
23. y =
ex – 1
ex + 1
: Ans. y0 = 2ex
(ex + 1)2 24. y = a2 exa – e- xa : Ans. y0 = 1 2 exa + e- xa
25. y = ecos x sin x: Ans. y0 = ecos x cos x – sin2 x
26. y = xnesin x. Ans. y0 = xn-1esin x(n + x cos x)
27. y = tg
1 – ex
1 + ex
: Ans. y0 = – 2ex
(1 + ex)2 ·
1
cos2 1 – ex
1 + ex
28. y = sin p1 – 2x . Ans. y0 = -cos p1 – 2x
2p1 – 2x 2x log 2
Exercise 2
Find the derivatives of the following functions: (try to use the logarithmic differentiation when it is possible):
1. y = s3 x(xx-2 + 1 1)2 . Ans. y0 = 1 3 s3 x(xx-2 + 1 1)2 x1 + x22+ 1 x – x -2 1
2. y =
(x + 1)3p4 (x – 2)3
p5 (x – 3)2 : Ans. y0 = (x + 1) p5 (3xp4-(x3)-2 2)3 × x + 1 3 + 4(x3- 2) – 5(x2- 3)
3. y =
(x + 1)2
(x + 2)3(x + 3)4 : Ans. y0 = -(x + 1) (x + 2) 5×42(+ 14 x + 3) x 5+ 5
4. y =
p5 (x – 1)2
p4 (x – 2)3p3 (x – 3)7 : Ans. y0 = 60p5 (x–161 1)3xp24 + 480 (x – 2) x -7p3271 (x – 3)10
5. y =
x 1 + x2
p1 – x2 : Ans. y0 = 1 + 3 (1 -x2x-2)23 2×4
6. y = x5(a + 3x)3(a – 2x)2. Ans. y0 = 5×4(a + 3x)2(a – 2x) a2 + 2ax – 12×2
7. y = arcsin
x a
: Ans. y0 = pa21- x2
8. y = (arcsin x)2: Ans. y0 = 2 arcsin p1 – x2x
9. y = cot x2 + 1 : Ans. y0 = 2x
1 + (x2 + 1)2 10. y = cot
2x
1 – x2 : Ans. y0 = 1 +2×2
11. y = arccos x2 : Ans. y0 = p1–2xx4 12. y = arccos x x: Ans. y0 = -(x + px21p-1 x-2xarccos 2 x)
13. y = arcsin
x + 1

2
14. y = xpa2 – x2 + a2 arcsin x
a
. Ans. y0 = 2pa2 – x2
15. y = pa2 – x2 + a arc sin x
a
. Ans. y0 = ra a – + x x
16. f(x) = arccos(log x): Ans. f0(x) = – 1
xp1 – log2 x
17. y = cot r1 1 + cos – cos x x(0 6 x < π): Ans. y0 = 1 2
18. y = cot
ex – e-x
2
: Ans. y0 = 2
ex + e-x
19. y = xarcsin x. Ans. y0 = xarcsin x arcsin x x + plog 1 -xx2
20. y = log 1 + 1 – xx
14
–
1 2
cot x: Ans. y0 = x2
1 – x4
21. y = log 1 + xp2 + x2
1 – xp2 + x2 + 2 cot
xp2
1 – x2 : Ans. y0 = 1 + 4px24
22. y = arccos
x2n – 1
x2n + 1: Ans. y0 = -x (2xn2jnx+ 1) jn
3

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