Differential Calculus 201-NYA-05
Vincent Carrier
Exercise Sheet 17
4.4 Derivative of the Other Trigonometric Functions
Find dy/dx.
√
tan x
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
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
Find the slope of the tangent line to the curve at the given value of x.
12. y =
tan x
sec x + tan x
13. y = csc2 x + cot2 x
x=π
x=−
5π
6
4.5 Inverse Trigonometric Functions
Find the following quantities.
14. arcsin (−1)
√ 15. arctan
3
18. arctan 1
√ !
3
19. arccos −
2
22. arccos 0
1
23. arcsin −
2
Find the derivative.
√
26. y = arctan x − 1
√
29. z = t arcsin t + 1 − t2
32. y = arcsin x arccos x
1
16. arccos −
2
1
20. arcsin √
2
24. arctan
1
√
3
27. z = arcsin2 t3
1
30. g(z) = arctan
z
√
33. f (x) = arcsin 3 x
17. arcsin
√ !
3
2
√ 21. arctan − 3
1
25. arccos − √
2
28. f (x) =
31. y =
√
3
1
arccos 3x
arccos x2
34. z = arctan3 t4
Find the equation of the tangent line to the curve at the given value of x.
x=−
35. y = arcsin 2x
1
4
36. y = arctan x2
Find the slope of the tangent line to the curve at the given value of x.
√
√
arcsin x
3
37. y =
x=
38. y = (arctan x)2
x
2
x=
√
4
3
x=1
Answers:
1.
dy
sec2 x
= √
dx
2 tan x
2.
dy
= 6 sec3 2x tan 2x
dx
3.
√
√
dy
csc x cot x
=−
√
dx
2 x
4.
dy
= −4x3 csc2 (x4 + 3)
dx
5.
dy
= 2 tan x sec2 x (sec2 x + tan2 x)
dx
6.
dy
sec
=
dx
7.
dy
= 4x csc2 (csc2 x2 ) csc2 x2 cot x2
dx
8.
dy
= sec x(tan x + x tan2 x + x sec2 x)
dx
9.
dy
1 − y sec2 xy
=
dx
x sec2 xy − 1
√
10.
dy
4
= 3 tan2 x sec2 x; y = x −
dx
3
11.
√
dy
= − csc x(cot2 x + csc2 x); y = 3 2x +
dx
14. −
20.
π
2
15.
π
4
3 + 10π
9
π
3
21. −
π
3
√
2(9π − 4)
4
16.
2π
3
17.
22.
π
2
23. −
dz
6t2 arcsin t3
= √
dt
1 − t6
26.
dy
1
√
=
dx
2x x − 1
27.
29.
dz
= arcsin t
dt
30. g 0 (z) = −
32.
dy
arccos x − arcsin x
√
=
dx
1 − x2
33. f 0 (x) =
1
p
3x2/3 1 − x2/3
√
dy
2
4
6−π 3
√
= √
; y = √ x+
dx
3
6 3
1 − 4x2
36.
35.
37.
√
tan x tan 3 tan x sec2 x
3(tan x)2/3
12.
sec x
dy
=
; −1
dx
(sec x + tan x)2
13.
√
dy
= −4 csc2 x cot x; −16 3
dx
π
3
π
6
18.
π
4
19.
5π
6
24.
π
6
25.
3π
4
28. f 0 (x) = √
1
1 + z2
√
√
dy
x − 1 − x2 arcsin x 12 3 − 4π
√
=
;
dx
9
x2 1 − x2
√
3
3
1 − 9x2 arccos2 3x
31.
dy
2x
√
=−
dx
3(arccos x2 )2/3 1 − x4
34.
dz
12t3 arctan2 t4
=
dt
1 + t8
√
√
4
dy
2x
3
2π − 3 3
=
;
y
=
x
+
dx
1 + x4
2
6
38.
√
dy
arctan x π
= √
;
dx
x(1 + x) 8