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Higher Order Derivatives Worksheet

Higher Order Derivatives Worksheet
  • Page 1
 1.  
Find f ″(x), if f(x) = 4x3 - 9x2 + 6.
a.
12x + 10
b.
24x - 18
c.
18 - 24x
d.
12x - 18


Solution:

f(x) = 4x3 - 9x2 + 6

f ′(x) = 12x2 - 18x
[Differentiate with respect to x.]

f ′′(x) = 24x - 18
[Differentiate f ′(x) with respect to x.]


Correct answer : (2)
 2.  
Find f ″(0), if f(x) = 4(x + 7)4.
a.
49
b.
- 48
c.
2352
d.
- 2352


Solution:

f(x) = 4(x + 7)4

f ′(x) = 16(x + 7)3(1) = 16(x + 7)3
[Differentiate with respect to x.]

f ′′(x) = 48(x + 7 )2
[Differentiate f ′(x) with respect to x.]

f ′′(0) = 48(0 + 7)2 = 2352
[Substitute x = 0.]


Correct answer : (3)
 3.  
A function f is defined by f(x) = 3e3x. Find f ″(- 3).
a.
- 1e9
b.
27e9
c.
1e3
d.
- 27e3


Solution:

f(x) = 3 e3x

f ′(x) = 9 e3x
[Differentiate with respect to x.]

f ″(x) = 27 e3x
[Differentiate f ′(x) with respect to x.]

f ″(- 3) = 27 e-9 = 27e9
[Substitute x = - 3.]


Correct answer : (2)
 4.  
A function f is defined by f(x) = 3ln |x|. What is f ′′(x)?
a.
1x2
b.
x-2
c.
- 3x2
d.
3x2


Solution:

f(x) = 3ln |x|

f ′(x) = 3 / x = 3x-1
[Differentiate with respect to x.]

f ′′(x) = - 3x-2 = - 3x2
[Differentiate f ′(x) with respect to x.]


Correct answer : (3)
 5.  
Find f ″(x), if f(x) = 4x24 + x.
a.
128x + 512(4 + x)3
b.
128x - 512(4 + x)4
c.
128x + 512(4 + x)4
d.
128x + 512(4 - x)4


Solution:

f(x) = 4x24 + x

f ′(x) = (4 + x)(8x) - 4x2(1)(4 + x)2
[Differentiate f ′(x) with respect to x by using quotient rule.]

= 32x + 4x2(4 + x)2

f ′′(x) = (4 + x)2(32 + 8x) -2 (4 + x)(4x2 + 32x)(4 + x)4
[Differentiate f ′(x) with respect to x by using quotient rule.]

= 128x + 512(4 + x)4


Correct answer : (3)
 6.  
Find f ″(x), if f(x) = x + 9.
a.
14(x + 9)32
b.
14(x + 9)12
c.
- 14(x + 9)32
d.
none of the above


Solution:

f(x) = x + 9 = (x + 9)1 / 2

f ′(x) = 1 / 2 (x + 9)-1 / 2
[Differentiate with respect to x.]

f ′′(x) = 1 / 2 (- 1 / 2 )(x + 9)- 3 / 2
[Differentiate f′(x) with respect to x.]

= - (x + 9)-324 or -14(x + 9)32


Correct answer : (3)
 7.  
What is the second derivative of the function f(x) = 7x65?
a.
- 42 25x- 6 5
b.
42 5x- 6 5
c.
42 25x- 4 5
d.
- 42 25x 6 5


Solution:

f(x) = 7x65

f ′(x) = 42 / 5x15
[Differentiate with respect to x.]

f ″(x) = 42 / 25x- 4 / 5
[Differentiate f ′(x) with respect to x.]


Correct answer : (3)
 8.  
A function f is defined by f(x) = 2x. Find f ″(7).
a.
- 2 - 7 (ln(2))2
b.
2 - 7 (ln(2))3
c.
2 - 7 (ln(2))2
d.
2 7 (ln(2))2


Solution:

f(x) = 2x

f ′(x) = 2xln(2)
[Differentiate with respect to x.]

f ′′(x) = 2x (ln(2))2
[Differentiate f ′(x) with respect to x.]

f ′′(7) = 27 (ln(2))2
[Substitute x = 7.]


Correct answer : (4)
 9.  
A function f is defined by f(x) = 6x ln |x|. What is f ″(x)?
a.
6x
b.
1x
c.
- 6x


Solution:

f(x) = 6x ln |x|

f ′(x) = (6)ln |x| + 6x (1x)
[Use product rule.]

= 6ln |x| + 6

f ′′(x) = 6 / x
[Differentiate f ′(x) with respect to x.]


Correct answer : (1)
 10.  
A function f is defined by f(x) = 5x4 + 8x3 - 5x2 + 6. Find f ″′(x).
a.
120x - 48
b.
120x + 48
c.
- 120x + 48
d.
48x - 120


Solution:

f(x) = 5x4 + 8x3 - 5x2 + 6

f ′(x) = 20x3 + 24x2 - 10x
[Differentiate f(x) with respect to x.]

f ′′(x) = 60x2 + 48x - 10
[Differentiate f ′(x) with respect to x.]

f ′′′(x) = 120x + 48
[Differentiate f ′′(x) with respect to x.]


Correct answer : (2)

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