Practice Problems 2
Name:
Find the derivative of the following functions:
1. f (x) = (3 − x2 )(x3 − x + 1)
2. f (x) = (x +
1
1
)(x − + 1)
x
x
3. f (x) = (5x3 − 2x2 +
√
1
x)(x8 + √ )
x
4. f (x) = 4x3 ex
5. f (x) = −2(x9 + 4x + 1 −
6. f (x) = (4ex + ln(x) − 4)(
7. f (x) =
2x + 5
3x − 2
8. f (x) =
x2 − 1
x2 + x − 2
√
x) ln(x)
1
1
1
−
+ 3)
x x2
x
√
x−1
9. f (x) = √
x+1
1
10. f (x) =
ex + 4x2 − 20
ln(x) + 3
11. f (x) = (x3 − 2x2 + 7x − 3)20
12. f (x) =
p
3
x2 − 4x + 16
1
13. f (x) = ln(9x2 − 4x + √ )
x
14. f (x) = 3xe4x+3
15. f (x) = (10x2 − 7x + 43)(2x2 − 3x + 8)2
16. f (x) =
3x + 2
2x + 1
3
17. f (x) = (2x + 5)2 e5x+1
r
18. f (x) =
ln(2x + 3)
3x2 + 1
2
Answer Key:
1. f 0 (x) = (−2x)(x3 − x + 1) + (3 − x2 )(3x2 − 1)
2. f 0 (x) = (1 − x−2 )(x −
1
x
+ 1) + (x + x1 )(1 + x−2 )
1
3. f 0 (x) = (15x2 − 4x + 12 x− 2 )(x8 +
√1 )
x
+ (5x3 − 2x2 +
√
3
x)(8x7 − 21 x− 2 )
4. f 0 (x) = 12x2 ex + 4x3 ex
1
5. f 0 (x) = −2(9x8 + 4 − 12 x− 2 ) ln x − 2(x9 + 4x + 1 −
6. f 0 (x) = (4ex + x1 )( x1 −
1
x2
+
1
x3 )
7. f 0 (x) =
2(3x−2)−3(2x+5)
(3x−2)2
8. f 0 (x) =
2x(x2 +2−2)−(x2 −1)(2x+1)
(x2 +x−2)2
9. f 0 (x) =
10. f 0 (x) =
1
1 −2
2x
√
x) x1
+ (4ex + ln x − 4)(−x−2 + 2x−3 − 3x−4 )
1
√
√
( x+1)−( x−1)( 12 x− 2 )
√
( x+1)2
1
(ex +8x)(ln x+3)−(ex +4x2 −20)( x
)
(ln x+3)2
11. f 0 (x) = 20(x3 − 2x2 + 7x − 3)19 (3x2 − 4x + 7)
2
12. f 0 (x) = 31 (x2 − 4x + 16)− 3 (2x − 4)
13. f 0 (x) =
3
· (18x − 4 − 12 x− 2 )
1
9x2 −4x+ √1x
14. f 0 (x) = 3e4x+3 + 3xe4x+3 · 4
15. f 0 (x) = (20x − 7)(2x2 − 3x + 8)2 + (10x2 − 7x + 43) · 2(2x2 − 3x + 8)(4x − 3)
16. f 0 (x) = 3
3x+2
2x+1
2
·
3(2x+1)−(3x+2)(2)
(2x+1)2
17. f 0 (x) = 2(2x + 5) · 2e5x+1 + (2x + 5)2 e5x+1 · 5
18. f 0 (x) =
1
2
ln(2x+3)
3x2 +1
− 12
·
2
1
2x+3 ·2(3x +1)−[ln(2x+3)](6x)
(3x2 +1)2
3