Differential Calculus 201-103-RE
Vincent Carrier
Exercise Sheet 14
4.3 Derivative of the Sine and Cosine Functions
Find the derivative of the following functions.
sin x
cos x
√
√
4. y = x cos x
1. y =
7. y = sin2
√
x
q
√
10. y = cos x
13. y =
2. y = cos x5
3. y = x4 sin x
5. y = sin3 2x
6. y = cos4 x3
sin x
8. y = √
cos x
9. y =
11. y =
sin2 x
1 + cos2 x
sin x
1 + sin2 x
cos x
14. y = √
1 + sin2 x
cos x
sin x − cos x
12. y = p
3
3
cos(sin x)
15. y = sin3 (cos4 x5 )
Find the equation of the tangent line to the curve at the given value of x.
16. y = sin2 x
x=−
√
18. y = sin x
x=
20. y =
sin x
sin x + cos x
2π
3
17. y =
4π 2
9
2
cos3 x
19. y = sin x cos x
x=
π
4
21. y = sin(2x) cos(3x)
Find the value(s) of x at which the tangent line is horizontal.
√
22. y = x + 2 sin x
23. y = x − cos 2x
25. y = cos2 x
x=
26. y = sin2 x cos2 x
3π
4
x=−
5π
6
x=
2π
3
24. y = sin x cos x
27. y = sin(π cos x)
Find dy/dx using implicit differentiation.
28. sin xy = x2 + y 2
29. sin3 x + cos3 y = x3 y 3
30.
p
sin x cos y = x + y
Answers:
dy
1
=
dx
cos2 x
1.
dy
cos
4.
=
dx
√
√
√
x − x sin x
√
2 x
2.
dy
= −5x4 sin x5
dx
3.
dy
= x3 (4 sin x + x cos x)
dx
5.
dy
= 6 sin2 (2x) cos(2x)
dx
6.
dy
= −12x2 cos3 (x3 ) sin(x3 )
dx
dy
1 + cos2 x
=
dx
2(cos x)3/2
9.
dy
1
=−
dx
(sin x − cos x)2
12.
dy
sin(sin x) cos x
=
dx
[cos(sin x)]4/3
7.
√
√
dy
sin( x) cos( x)
√
=
dx
x
8.
10.
√
dy
sin x
=− √ p
√
dx
4 x cos x
11.
dy
cos3 x
=
dx
(1 + sin2 x)2
13.
dy
4 sin x cos x
=
dx
(1 + cos2 x)2
14.
dy
2 sin x
=−
dx
(1 + sin2 x)3/2
15.
dy
= −60x4 sin2 (cos4 x5 ) cos(cos4 x5 ) cos3 (x5 ) sin(x5 )
dx
√
√
9 + 4π 3
3
dy
= 2 cos x sin x; y =
x+
16.
dx
2
12
17.
√
√
dy
6 sin x
=
; y = 12 2x − 2(4 + 9π)
4
dx
cos x
18.
√
√
cos x
3
3 3+π
dy
= √ ; y =− x+
dx
8π
6
2 x
19.
√
dy
1
3 3 + 5π
= cos2 x − sin2 x; y = x +
dx
2
12
20.
dy
1
1
4−π
=
; y = x+
2
dx
(sin x + cos x)
2
8
21.
√
4π − 3 3
dy
= 2 cos(2x) cos(3x) − 3 sin(2x) sin(3x); y = −x +
dx
6
28.
22.
√
dy
3π
5π
= 1 + 2 cos x; x =
+ 2πk,
+ 2πk, k ∈ Z
dx
4
4
23.
7π
11π
dy
= 1 + 2 sin 2x; x =
+ πk,
+ πk, k ∈ Z
dx
12
12
24.
dy
π π
= cos2 x − sin2 x; x = + k, k ∈ Z
dx
4
2
25.
dy
π
= −2 cos x sin x; x = k, k ∈ Z
dx
2
26.
dy
π
= 2 sin x cos x(cos2 x − sin2 x); x = k, k ∈ Z
dx
4
27.
dy
π
= −π cos(π cos x) sin x; x = k, k ∈ Z
dx
3
dy
y cos(xy) − 2x
=
dx
2y − x cos(xy)
29.
√
cos x cos y − 2 sin x cos y
dy
√
30.
=
dx
sin x sin y + 2 sin x cos y
dy
sin2 x cos x − x2 y 3
=
dx
sin y cos2 y + x3 y 2