# Higher Order Derivatives Worksheet - Page 2

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

#### 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.]

12.
Find $f$ ′′($x$), if $f$($x$) = 2$x$3 + 7$x$2 - 3$x$ + 9.
 a. 14$x$ - 12 b. 14$x$2 + 12 c. 12$x$ + 14 d. 12$x$

#### 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.]

13.
If $f$ ′($x$) = 2$x$ - 3$x$2, 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.]

14.
If $f$ ′($x$) = 8$x$2 + 5$x$- 4 then $f$ ′′′($x$) = ?
 a. 16 + 100$x$-6 b. 16 - 100$x$ c. 100 - 16$x$-6 d. 16 - 20$x$-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.]

15.
If a function $f$ is defined by $f$($x$) = , then
 a. $f$ ′′($x$) = b. $f$ ′′($x$) = c. $f$ ′′($x$) = d. $f$ ′′($x$) =

#### 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.]

16.
A function $f$ is defined by $f$($x$) = 4$x$4 + 2$x$3 - 3$x$2 + 2$x$ - 4. Find $f$ ″($x$).
 a. 48$x$ + 12 b. 12$x$2 - 48$x$ + 6 c. 12$x$2 + 48$x$ d. 48$x$2 + 12$x$ - 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.]

17.
A function 'R' is defined by R($x$) = 90000 - 4$x$3 + 8$x$2 + 400$x$. What is R″($x$)?
 a. - 24$x$ - 16 b. - 16$x$ + 24 c. 16$x$ - 24 d. - 24$x$ + 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.]

18.
If a function $f$ is defined by $f$($x$) = - 5$x$3 - 9$x$2 - 9$x$ + 5, then what is $f$ ″($x$)?
 a. - 18$x$ + 30 b. 30$x$ + 18 c. - 30$x$ - 18 d. 30$x$ - 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.]

19.
If $f$ ′($x$) = 9$x$2 + 2$x$ + 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.]

20.
If $y$ = 2$x$ - 3$x$4, then find $\frac{{d}^{4}y}{d{x}^{4}}$.
 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.]