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Higher Order Derivatives Worksheet - Page 2

Higher Order Derivatives Worksheet
  • Page 2
 11.  
A function f is defined by f(x) = x - 5x. Find f 4(x).
a.
120x5
b.
- 120x5
c.
- 120x5
d.
120x5


Solution:

f(x) = x - 5x

f ′(x) = x(1) - (x - 5)(1)x2 = 5x2 = 5x-2
[Differentiate with respect to x.]

f ′′(x) = - 10 x-3
[Differentiate f ′(x) with respect to x.]

f ″′(x) = 30 x- 4
[Differentiate f′′(x) with respect to x.]

f 4 (x) = - 120 x-5 = - 120x5
[Differentiate f ″′(x) with respect to x.]


Correct answer : (2)
 12.  
Find f ′′(x), if f(x) = 2x3 + 7x2 - 3x + 9.
a.
14x - 12
b.
14x2 + 12
c.
12x + 14
d.
12x


Solution:

f(x) = 2x3 + 7x2 - 3x + 9

f ′(x) = 6x2 + 14x - 3
[Differentiate f(x) with respect to x.]

f ′′(x) = 12x + 14
[Differentiate f ′(x) with respect to x.]


Correct answer : (3)
 13.  
If f ′(x) = 2x - 3x2, then f ′′′(x) = ?
a.
3
b.
6
c.
- 6


Solution:

f ′(x) = 2x - 3x2

f ′′(x) = 2 - 6x
[Differentiate f ′(x) with respect to x.]

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


Correct answer : (3)
 14.  
If f ′(x) = 8x2 + 5x- 4 then f ′′′(x) = ?
a.
16 + 100x-6
b.
16 - 100x
c.
100 - 16x-6
d.
16 - 20x-6


Solution:

f ′(x) = 8x2 + 5x- 4

f ′′(x) = 16x - 20x-5
[Differentiate f ′(x) with respect to x.]

f ′′′(x) = 16 + 100x- 6
[Differentiate f ′′(x) with respect to x.]


Correct answer : (1)
 15.  
If a function f is defined by f(x) = 2x - 3 , then
a.
f ′′(x) = 4(x - 3)3
b.
f ′′(x) = 4(x + 3)3
c.
f ′′(x) = 4(x - 3)2
d.
f ′′(x) = - 4(x - 3)3


Solution:

f(x) = 2x - 3

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

f ′′(x) = 4(x - 3)3
[Differentiate f ′(x) with respect to x.]


Correct answer : (1)
 16.  
A function f is defined by f(x) = 4x4 + 2x3 - 3x2 + 2x - 4. Find f ″(x).
a.
48x + 12
b.
12x2 - 48x + 6
c.
12x2 + 48x
d.
48x2 + 12x - 6


Solution:

f(x) = 4x4 + 2x3 - 3x2 + 2x - 4

f ′(x) = 16x3 + 6x2 - 6x + 2
[Differentiate f(x) with respect to x.]

f ′′(x) = 48x2 + 12x - 6
[Differentiate f ′(x) with respect to x.]


Correct answer : (4)
 17.  
A function 'R' is defined by R(x) = 90000 - 4x3 + 8x2 + 400x. What is R″(x)?
a.
- 24x - 16
b.
- 16x + 24
c.
16x - 24
d.
- 24x + 16


Solution:

R(x) = 90000 - 4x3 + 8x2 + 400x

R′(x) = - 12x2 + 16x + 400
[Differentiate R(x) with respect to x.]

R″(x) = - 24x + 16
[Differentiate R′(x) with respect to x.]


Correct answer : (4)
 18.  
If a function f is defined by f(x) = - 5x3 - 9x2 - 9x + 5, then what is f ″(x)?
a.
- 18x + 30
b.
30x + 18
c.
- 30x - 18
d.
30x - 18


Solution:

f(x) = - 5x3 - 9x2 - 9x + 5

f ′(x) = - 15x2 - 18x - 9
[Differentiate f(x) with respect to x.]

f ′′(x) = -30x - 18
[Differentiate f ′(x) with respect to x.]


Correct answer : (3)
 19.  
If f ′(x) = 9x2 + 2x + 2, then f ′′′(x) = ?
a.
18
b.
21
c.
20
d.
19


Solution:

f ′(x) = 9x2 + 2x + 2

f ′′(x) = 18x + 2
[Differentiate f ′(x) with respect to x.]

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


Correct answer : (1)
 20.  
If y = 2x - 3x4, then find d4ydx4.
a.
72
b.
- 36
c.
- 72
d.
12


Solution:

y = 2x - 3x4

dydx = ddx(2x - 3x4) = 2 - 12x3
[Differentiate with respect to x.]

d2ydx2 = ddx(2 - 12x3) = - 36 x2
[Differentiate with respect to x.]

d3ydx3 = ddx(- 36 x2) = - 72x
[Differentiate with respect to x.]

d4ydx4 = ddx(- 72x) = - 72
[Differentiate with respect to x.]


Correct answer : (3)

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