﻿ Higher Order Derivatives Worksheet - Page 3 | Problems & Solutions Higher Order Derivatives Worksheet - Page 3

Higher Order Derivatives Worksheet
• Page 3
21.
If $\frac{dy}{dx}$ = - 6$x$ - 3 $x$2, then find $\frac{{d}^{3}y}{d{x}^{3}}$. 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.]

22.
If $f$($x$) = 10 $\sqrt{x}$, then find $f$ ″′($x$). a. $\frac{15}{4}$ b. - $\frac{15}{4}$ c. 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.]

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)

24.
If $f$($x$) = 5tan-1 ($x$), then $f$ ″′(1) = ? a. $\frac{5}{2}$ b. 40 c. - 40 d. - $\frac{5}{2}$

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.]

25.
If $f$($x$) = 3tan-1 ($x$), then $f$ ′′′($x$) = ? a. $\frac{18{x}^{4}-{x}^{2}-6}{1+{x}^{2}}$ b. $\frac{18{x}^{4}+12{x}^{2}-6}{{\left(1+{x}^{2}\right)}^{4}}$ c. $\frac{18{x}^{4}+12{x}^{2}}{1+{x}^{2}}$ d. $\frac{{x}^{4}+12{x}^{2}-6}{{x}^{2}}$

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.]