Differential Calculus 201-103-RE
Vincent Carrier
Exercise Sheet 15
4.4 Derivative of the Other Trigonometric Functions
Find dy/dx.
√
4. y = cot(x4 + 3)
5. y = tan2 x sec2 x
√
3. y = csc x
√
3
6. y = sec tan x
7. y = cot(csc2 x2 )
8. y = x tan x sec x
9. tan xy = x + y
1. y =
2. y = sec3 2x
tan x
Find the equation of the tangent line to the curve at the given value of x.
10. y = tan3 x
x=
5π
6
x=−
11. y = csc x cot x
3π
4
4.5 Inverse Trigonometric Functions
Find the following quantities.
12. arcsin (−1)
16. arctan 1
20. arccos 0
√ 13. arctan
3
√ !
3
17. arccos −
2
1
21. arcsin −
2
1
14. arccos −
2
√ !
2
18. arcsin
2
1
22. arctan √
3
15. arcsin
√ !
3
2
√ 19. arctan − 3
√ !
2
23. arccos −
2
Find the derivative.
√
24. y = arctan x − 1
27. z = t arcsin t +
√
1 − t2
30. y = arcsin x arccos x
1
arccos 3x
1
29. g(z) = arctan
z
25. z = arcsin2 t3
28. y =
√
3
26. f (x) =
arccos x2
31. f (x) = arcsin
√
3
x
32. z = arctan3 t4
Find the equation of the tangent line to the curve at the given value of x.
33. y = arcsin 2x
x=−
1
4
34. y = arctan x2
x=
√
4
3
Answers:
dy
sec2 x
= √
dx
2 tan x
√
√
dy
csc x cot x
√
3.
=−
dx
2 x
1.
dy
5.
= 2 tan x sec2 x (sec2 x + tan2 x)
dx
7.
dy
= 4x csc2 (csc2 x2 ) csc2 x2 cot x2
dx
9.
1 − y sec2 xy
dy
=
dx
x sec2 xy − 1
2.
dy
= 6 sec3 2x tan 2x
dx
4.
dy
= −4x3 csc2 (x4 + 3)
dx
dy
sec
6.
=
dx
8.
dy
4
10.
= 3 tan2 x sec2 x; y = x −
dx
3
√
3
dy
= sec x(tan x + x tan2 x + x sec2 x)
dx
√
3 + 10π
9
√
dy
11.
= − csc x(cot2 x + csc2 x); y = 3 2x +
dx
12. −
24.
π
2
√
2(9π − 4)
4
13.
π
3
14.
2π
3
15.
5π
6
18.
π
4
19. −
π
6
22.
π
6
23.
16.
π
4
17.
20.
π
2
21. −
dy
1
= √
dx
2x x − 1
26. f 0 (x) = √
√
tan x tan 3 tan x sec2 x
3(tan x)2/3
3
1 − 9x2 arccos2 3x
π
3
3π
4
25.
dz
6t2 arcsin t3
= √
dt
1 − t6
27.
dz
= arcsin t
dt
28.
dy
2x
√
=−
2
dx
3(arccos x )2/3 1 − x4
29. g 0 (z) = −
30.
arccos x − arcsin x
dy
√
=
dx
1 − x2
31. f 0 (x) =
32.
dz
12t3 arctan2 t4
=
dt
1 + t8
√
dy
2
4
6−π 3
√
√
√
33.
=
; y=
x+
dx
3
6 3
1 − 4x2
π
3
1
1 + z2
3x2/3
1
√
1 − x2/3
√
√
4
dy
2x
3
2π − 3 3
34.
=
; y=
x+
dx
1 + x4
2
6