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Higher Order Derivatives Worksheet - Page 3

Higher Order Derivatives Worksheet
  • Page 3
 21.  
If dydx = - 6x - 3 x2, then find d3ydx3.
a.
- 6
b.
6
c.
3


Solution:

dydx = - 6x - 3 x2

d2ydx2 = ddx(- 6x - 3 x2)
[Differentiate with respect to x.]

d2ydx2 = - 6 - 6x

d3ydx3 = ddx(- 6 - 6x) = - 6
[Differentiate with respect to x.]


Correct answer : (1)
 22.  
If f(x) = 10 x, then find f ″′(x).
a.
15 4x- 52
b.
- 15 4x- 52
c.
x- 52
d.
none of the above


Solution:

f(x) = 10 x = 10 x12

f ′(x) = 5 x- 12
[Differentiate f(x) with respect to x.]

f ′′(x) = - 5 / 2x- 32
[Differentiate f ′(x) with respect to x.]

f ′′′(x) = 15 / 4x- 52
[Differentiate f ′′(x) with respect to x.]


Correct answer : (1)
 23.  
If f(x) = 3sin x, then which of the following is true?
a.
f 4 (x) = f(x)
b.
f ′′′(x) = f(x)
c.
f ′′′(x) = - f(x)
d.
none of the above


Solution:

f(x) = 3Sin x

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

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

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

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

So, f 4(x) = f(x)


Correct answer : (1)
 24.  
If f(x) = 5tan-1 (x), then f ″′(1) = ?
a.
5 2
b.
40
c.
- 40
d.
- 5 2


Solution:

f(x) = 5tan-1x

f ′(x) = 51+x2
[Differentiate with respect to x.]

f ′′(x) = - 10x(1+x2)2
[Use Quotient rule.]

f ′′′(x) = 30x4 + 20x2 - 10(1+x2)4
[Use Quotient rule.]

f ′′′(1) = 5 / 2
[Substitute x = 1.]


Correct answer : (1)
 25.  
If f(x) = 3tan-1 (x), then f ′′′(x) = ?
a.
18x4-x2-61+x2
b.
18x4+12x2-6(1+x2)4
c.
18x4+12x21+x2
d.
x4+12x2-6x2


Solution:

f(x) = 3tan-1(x)

f ′(x) = 31+x2
[Differentiate f(x) with respect to x by using quotient rule.]

f ′′(x) = - 6x(1+x2)2
[Use quotient rule.]

f ′′′(x) = 18x4 + 12x2 - 6(1+x2)4
[Use quotient rule.]


Correct answer : (2)

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