Practice/Review of derivitive
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the derivative.
1) f(x) = 3x2 - 4x - 1, find f'(x)
A) 6x - 4
1)
C) 3x2 - 4
B) 3x - 4
2) y = 13x-2 + 7x3 + 1x, find f'(x)
A) -26x-3 + 21x2 + 1
2)
B) -26x-3 + 21x2
C) -26x-1 + 21x2 + 1
D) -26x-1 + 21x2
3) f(x) = 9x7/5 - 5x2 + 104 , find f'(x)
63 6/5
A)
x
- 10x
5
C)
3)
63 2/5
B)
x
- 10x + 4000
5
63 2/5
x
- 10x
5
D)
4) g(x) = 3x5 + x4 - 2x2 + 4, find g'(-1)
A) 11
B) 0
5) f(x) =
A)
D) 6x2 - 4
63 6/5
x
- 10x + 4000
5
4)
C) 15
D) 19
x+7
, find f'(x)
x
1
2 x
-
7
2 x3/2
5)
B) x3/2 + 7 x
C)
1
7
+
3/2
x
x
D)
1
2 x
-
7
2x
Find the derivative of the given function.
6) y = (x2 + 2)3
6)
A) 6x5 + 24x3 + 24x
C) 6x5 + 12x3 + 12x
B) 3x5 + 24x3 + 24x
D) 6x5 + 20x3 + 24x
Use the product rule to find the derivative.
7) f(x) = (3x - 4)(5x3 - x2 + 1)
7)
A) f'(x) = 60x3 - 69x2 + 8x + 3
B) f'(x) = 45x3 + 69x2 - 23x + 3
C) f'(x) = 60x3 - 23x2 + 69x + 3
D) f'(x) = 15x3 + 23x2 - 69x + 3
8) f(x) = (5x + 6)2
A) f'(x) = 10x + 12
8)
B) f'(x) = 50x + 60
C) f'(x) = 25x + 36
9) f(x) = (5x - 5)( x + 2)
A) f'(x) = 7.5x1/2 - 5x-1/2 + 10
D) f'(x) = 25x + 30
9)
B) f'(x) = 7.5x1/2 - 2.5x-1/2 + 10
C) f'(x) = 3.33x1/2 - 5x-1/2 + 10
D) f'(x) = 3.33x1/2 - 2.5x-1/2 + 10
1
10) g(x) = (x-5 + 3)(x-3 + 5)
A) g'(x) = -8x-9 - 25x-6 - 9x-4
10)
B) g'(x) = -8x-9 - 25x-4 - 9x-4
D) g'(x) = -8x-9 - 25x-6 - 9x-2
C) g'(x) = -8x-7 - 25x-6 - 9x-4
Use the quotient rule to find the derivative.
1
11) f(x) =
7
x +2
11)
A) f'(x) =
1
7
(7x + 2)2
B) f'(x) = -
C) f'(x) =
7x6
(x7 + 2)2
D) f'(x) = -
12) y =
7x6
7
(x + 2)2
1
(7x 7 + 2)2
x2 - 4
x
A)
dy
4
=x+
dx
x2
12)
B)
dy
4
=1+
dx
x
C)
dy
4
=1dx
x2
D)
dy
4
=1+
dx
x2
Find the derivative.
13) y = 4e9x
A) 36e9x
13)
B) 4xe36x
C) 4e36x
B) 10xe + 1
C) 10xex2 + 1
D) 36xe9x
14) y = e5x2 + x
A) 10xe5x2 + 1
14)
15) y = (x + 8)5 e-2x
A) -(x + 8)4 (2x + 11) e-2x
15)
B) -10(x + 8)4 e-2x
D) (x + 8)4 (x + 13) e-2x
C) -(x + 8)4 (2x + 11) e-3x
16) y = 8 -x
A) 8 -x
D) 10xe2x + 1
16)
B) -8 -x
C) - ln 8 (8 -x)
D) ln 8 (8 -x)
B) 4 (ln 5) 5 4x
C) 20 (ln 5) 5 4x
D) 20 (ln 4) 5 4x
17) y = 5 4x
A) 5 (ln 4) 5 4x
17)
18) y = 15x - 1
A) 15x - 1 ln x
18)
B) 15 ln 15
D) 15x - 1 ln 15x - 1
C) 15x - 1 ln 15
19) y = 9 x2
A) 9 x2 x ln 9
19)
C) 9 x2 2x ln 9
B) 2x ln 9
2
D) 9 x2 2x ln x
Find the derivative of the function.
20) y = ln 2x
1
A) 2x
21) y = ln 4x2
2x
A)
2
x +4
20)
1
B) x
1
C)
2x
1
D)
x
2
B)
x
8
C)
x
1
D)
2x + 4
21)
22) y = ln (x - 4)
1
A)
4-x
22)
B) -
1
x+4
C)
1
x-4
D)
1
x+4
Find the derivative.
23) y = ex ln x, x > 0
23)
A) ex ln x
B)
ex(ln x + x)
x
C)
24) y = ex5 ln x
ex5 + 5x4 ex5 ln x
A)
x
C)
ex
x
D)
ex(x ln x + 1)
x
24)
ex5 + 5x5 ex5 ln x
x
B)
5x5 ex5 + 1
x
D)
ex5 + 5ex5 ln x
x
Identify the open intervals where the function is changing as requested.
25) Increasing
25)
f(x)
3
2
1
-3
-2
-1
1
2
3
x
-1
-2
-3
A) (-2, ∞)
B) (-2, 2)
C) (-3, ∞)
3
D) (-3, 3)
26) Decreasing
26)
f(x)
3
2
1
-3
-2
-1
1
2
x
3
-1
-2
-3
A) (0, -2)
B) (-∞, -2)
C) (-∞, -3)
D) (-3, -2)
27) Increasing
27)
f(x)
3
2
1
-6 -5 -4 -3 -2 -1
1
2
3
4
5
6 x
-1
-2
-3
A) (-3, 0), (3, ∞)
B) (0, 3)
C) (-∞, -3), (0, 3)
D) (-∞, -3), (3, ∞)
28) Decreasing
28)
f(x)
3
2
1
-6 -5 -4 -3 -2 -1
1
2
3
4
5
6 x
-1
-2
-3
A) (-3, 3)
B) (-3, 0), (3, ∞)
C) (-∞, -3), (0, 3)
4
D) (-3, 0)
Suppose that the function with the given graph is not f(x), but f′(x). Find the open intervals where f(x) is increasing or
decreasing as indicated.
29) Increasing
29)
y
5
4
3
2
1
-5
-4
-3
-2
-1
1
2
3
4
5
x
-1
-2
-3
-4
-5
A) (2, ∞)
B) (0, ∞)
C) (-2, 2)
D) (-∞, -2), (2, ∞)
30) Decreasing
30)
5
y
4
3
2
1
-5
-4
-3
-2
-1
1
2
3
4
5 x
-1
-2
-3
-4
-5
A) (0, ∞)
B) (-1, 1)
C) (-∞, 0)
D) (-∞, -1), (1, ∞)
Find all the critical numbers of the function.
31) f(x) = xe-2x
1
A)
2
31)
B) e-2x
C) 0
5
D) -2
Find the open intervals where the function is concave upward or concave downward. Find any inflection points.
32)
32)
A) Concave upward on (-1, ∞); concave downward on (-∞, 2); inflection points at (-1, 0) and (2,
-3)
B) Concave upward on (0, ∞); concave downward on (-∞, 0); inflection points at (-4, 0), (-1, 0),
7
and , 0
2
C) Concave upward on (0, ∞); concave downward on (-∞, 0); inflection point at (0, -1)
D) Concave upward on (-1, ∞); concave downward on (-∞, 2); inflection point at (2, -3)
33)
33)
A) Concave upward on (-∞, -1) and (2, ∞); concave downward on (-1, 2); inflection point at (0,
-1)
B) Concave upward on (-∞, -1) and (1, ∞); concave downward on (-1, 1); inflection points at (-1,
-3) and (1, -2)
C) Concave upward on (-∞, -3) and (2, ∞); concave downward on (-3, 2); inflection points at (-1,
-3) and (1, -2)
D) Concave upward on (-∞, -1) and (1, ∞); concave downward on (-1, 1); inflection points at (-3,
-5), (0, -1), and (2, -2)
6
Answer Key
Testname: UNTITLED1
1) A
2) A
3) C
4) C
5) A
6) A
7) A
8) B
9) B
10) A
11) B
12) D
13) A
14) A
15) A
16) C
17) B
18) C
19) C
20) D
21) B
22) C
23) D
24) C
25) B
26) D
27) C
28) B
29) D
30) D
31) A
32) C
33) B
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