Differential Calculus 201-NYA-05
Vincent Carrier
Exercise Sheet 11
3.5 Product and Quotient Rules
Find the derivative of the following functions.
1. f (x) =
x
3x + 2
2. y =
3 − 2x2
4. y =
2 − 3x2
√
t
7. z = √
t+1
4 − 3x
5 − 2x
3. z =
3t6
7
√
3x − 4 x
√
6. f (x) =
3
x
2t3
5. g(t) =
3 − t3
√
x+ x
√
8. y =
x− x
x
9. g(x) =
x+
1
x
3.6 Chain Rule
Find the derivative of the following functions.
10. y = (2x2 + 3)9
13. y =
11. f (x) =
3
2
(x + 2x + 3)4
16. g(t) = t3 (t + 3)6
12. y =
√
3
14. f (x) = 9 4 − x2
20. f (x) =
t3
(2t + 1)4
x+1
x−1
√
x3 − 2
15. y = √
4
8
x5 + 6
√ √
18. f (x) = x( x + 2)8
17. y = x6 (x5 − 4)3
√
19. y = x3 x4 + 8
22. z =
1
(5 − 3x)8
5
r
21. z =
t2 + 4
t2 + 9
x2
24. y = √
x2 + 9
1−z
23. g(z) = √
3 − 2z
25. f (x) = (2x + 1)4 (2 − 3x)3
26. y = (x2 + 1)5 (x2 − 1)4
27. y = [x3 + (4x + 3)8 ]9
28. g(t) =
r
29. f (x) =
q
√
x+ x+ x
p
4t2 + (3 − 4t3 )5
r
30. y =
4
x3
q
√
3
+ x2 + x
Answers:
1. f 0 (x) =
4.
2
(3x + 2)2
2.
10x
dy
=
dx
(2 − 3x2 )2
dy
7
=−
dx
(5 − 2x)2
5. g 0 (t) =
18t2
(3 − t3 )2
10.
dy
= 36x(2x2 + 3)8
dx
11. f 0 (x) =
13.
dy
24(x + 1)
=− 2
dx
(x + 2x + 3)5
14. f 0 (x) = −
17.
9. g 0 (x) =
24
(5 − 3x)9
6x
(4 − x2 )2/3
dy
= 3x5 (7x5 − 8)(x5 − 4)2
dx
19.
dy
x2 (5x4 + 24)
= √
dx
x4 + 8
20. f 0 (x) = −
22.
dz
t2 (3 − 2t)
=
dt
(2t + 1)5
23. g 0 (z) =
25. f 0 (x) = 7(1 − 6x)(2x + 1)3 (2 − 3x)2
10(x + 1)4
(x − 1)6
z−2
(3 − 2z)3/2
26.
dz
18t5
=
dt
7
6. f 0 (x) =
√
dy
x
√ 2
8.
=−
dx
(x − x)
dz
1
7.
= √ √
dt
2 t( t + 1)2
16. g 0 (t) = 9t2 (t + 1)(t + 3)5
3.
2
x1/3
(x2
−
2x
+ 1)2
12.
dy
3x2
= √
dx
2 x3 − 2
15.
dy
10x4
=− 5
dx
(x + 6)5/4
18. f 0 (x) =
√
√
(9 x + 2)( x + 2)7
√
2 x
21.
dz
5t
√
=
2
3/2
dt
(t + 9)
t2 + 4
24.
dy
x(x2 + 18)
= 2
dx
(x + 9)3/2
dy
= 2x(9x2 − 1)(x2 + 1)4 (x2 − 1)3
dx
2t[2 − 15t(3 − 4t3 )4 ]
dy
= 9[x3 + (4x + 3)8 ]8 [3x2 + 32(4x + 3)7 ]
28. g 0 (t) = p
dx
4t2 + (3 − 4t3 )5
"
#
1
1
1
0
p
√
29. f (x) = q
p
√ 1 + 2 x + √x 1 + 2 x
2 x+ x+ x
27.
dy
30.
=
dx
1
p
√ 3/4
4 x3 + 3 x2 + x
"
1
3x +
√ 2/3
2
3 (x + x)
2
2
3x5/6
1
2x + √
2 x
#