Higher Order Derivatives Worksheet

Higher Order Derivatives Worksheet
• Page 1
1.
Find $f$ ″($x$), if $f$($x$) = 4$x$3 - 9$x$2 + 6.
 a. 12$x$ + 10 b. 24$x$ - 18 c. 18 - 24$x$ d. 12$x$ - 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.]

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.]

3.
A function $f$ is defined by $f$($x$) = 3$e$3x. Find $f$ ″(- 3).
 a. - $\frac{1}{{e}^{9}}$ b. $\frac{27}{{e}^{9}}$ c. $\frac{1}{{e}^{3}}$ d. - $\frac{27}{{e}^{3}}$

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.]

4.
A function $f$ is defined by $f$($x$) = 3ln |$x$|. What is $f$ ′′($x$)?
 a. $\frac{1}{{x}^{2}}$ b. $x$-2 c. - $\frac{3}{{x}^{2}}$ d. 3$x$2

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.]

5.
Find $f$ ″($x$), if $f$($x$) = .
 a. b. c. d.

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

6.
Find $f$ ″($x$), if $f$($x$) = .
 a. b. c. 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

7.
What is the second derivative of the function $f$($x$) = 7${x}^{\frac{6}{5}}$?
 a. - $\frac{42}{25}$$x$- $\frac{6}{5}$ b. $\frac{42}{5}$$x$- $\frac{6}{5}$ c. $\frac{42}{25}$$x$- $\frac{4}{5}$ d. - $\frac{42}{25}$$x$ $\frac{6}{5}$

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.]

8.
A function $f$ is defined by $f$($x$) = 2x. Find $f$ ″(7).
 a. - 2 - 7 ${\left(\mathrm{ln}\left(2\right)\right)}^{2}$ b. 2 - 7 ${\left(\mathrm{ln}\left(2\right)\right)}^{3}$ c. 2 - 7 ${\left(\mathrm{ln}\left(2\right)\right)}^{2}$ d. 2 7 ${\left(\mathrm{ln}\left(2\right)\right)}^{2}$

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.]

9.
A function $f$ is defined by $f$($x$) = 6$x$ ln |$x$|. What is $f$ ″($x$)?
 a. $\frac{6}{x}$ b. $\frac{1}{x}$ c. - $\frac{6}{x}$

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.]

10.
A function $f$ is defined by $f$($x$) = 5$x$4 + 8$x$3 - 5$x$2 + 6. Find $f$ ″′($x$).
 a. 120$x$ - 48 b. 120$x$ + 48 c. - 120$x$ + 48 d. 48$x$ - 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.]