# Calculate the derivatives

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

p2 : Ans. y0 = | p1 – 21x – x2 |

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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