Mathematics Questions and Answers – Second Order Derivatives

«
»

This set of Mathematics Multiple Choice Questions & Answers (MCQs) focuses on “Second Order Derivatives”.

1. Find the second order derivative of y=9 log⁡ t3.
a) \(\frac{27}{t^2}\)
b) –\(\frac{27}{t^2}\)
c) –\(\frac{1}{t^2}\)
d) –\(\frac{27}{2t^2}\)
View Answer

Answer: b
Explanation: Given that, y=9 log⁡t3
\(\frac{dy}{dx}=9.\frac{1}{t^3}.3t^2=\frac{27}{t}\)
\(\frac{d^2 y}{dx^2}=27(-\frac{1}{t^2})=-\frac{27}{t^2}\).
advertisement

2. Find \(\frac{d^2y}{dx^2}\), if y=tan2⁡x+3 tan⁡x.
a) sec2⁡⁡x tan⁡x (2 tan⁡x+sec⁡x+3)
b) 2 sec2⁡⁡x tan⁡x (2 tan⁡x-sec⁡x+3)
c) 2 sec2⁡⁡x tan⁡x (2 tan⁡x+sec⁡x+3)
d) 2 sec2⁡⁡x tan⁡x (2 tan⁡x+sec⁡x-3)
View Answer

Answer: c
Explanation: Given that, y=tan2⁡⁡x+3 tan⁡x
\(\frac{dy}{dx}\)=2 tan⁡x sec2⁡⁡x+3 sec2⁡x=sec2⁡⁡x (2 tan⁡x+3)
By using the u.v rule, we get
\(\frac{d^2 y}{dx^2}=\frac{d}{dx}\) (sec2⁡⁡x).(2 tan⁡x+3)+\(\frac{d}{dx}\) (2 tan⁡x+3).sec2⁡⁡x
\(\frac{d^2 y}{dx^2}\)=2 sec2⁡⁡x tan⁡x (2 tan⁡x+3)+sec2⁡⁡x (2 sec⁡x tanx)
=2 sec2⁡x tan⁡x (2 tan⁡x+sec⁡x+3).

3. If y=6x2+3, then \(\left (\frac{dy}{dx}\right )^2=\frac{d^2 y}{dx^2}\).
a) True
b) False
View Answer

Answer: b
Explanation: The given statement is false. Given that, y=6x2+3
\(\frac{dy}{dx}\)=12x
⇒\(\left (\frac{dy}{dx}\right )^2=(12x)^2=144x^2\)
\(\frac{d^2 y}{dx^2}=\frac{d}{dx}\) (12x)=12
∴\(\left (\frac{dy}{dx}\right )^2≠\frac{d^2 y}{dx^2}\)
advertisement
advertisement

4. Find the second order derivative of y=2e2x-3 log⁡(2x-3).
a) 8e2x+\(\frac{1}{(2x-3)^2}\)
b) 8e2x–\(\frac{12}{(2x-3)^2}\)
c) e2x+\(\frac{12}{(2x-3)^2}\)
d) 8e2x+\(\frac{12}{(2x-3)^2}\)
View Answer

Answer: d
Explanation: Given that, y=2e2x-3 log⁡(2x-3)
\(\frac{dy}{dx}\)=4e2x-3.\(\frac{1}{(2x-3)}\).2=4e2x–\(\frac{6}{(2x-3)}\)
\(\frac{d^2 y}{dx^2}=\frac{d}{dx} (\frac{dy}{dx})\)=8e2x+\(\frac{12}{(2x-3)^2}\)

5. Find \(\frac{d^2 y}{dx^2}\), if y=2 sin-1⁡(cos⁡x).
a) 0
b) sin-1\((\frac{1}{cos⁡x})\)
c) 1
d) -1
View Answer

Answer: a
Explanation: Given that, y=2 sin-1⁡(cos⁡x)
\(\frac{dy}{dx}=2.\frac{1}{\sqrt{1-cos^2⁡x}}\).-sin⁡x=-2 (∵\(\sqrt{1-cos^2⁡x}\)=sin⁡x)
\(\frac{d^2 y}{dx^2}\)=\(\frac{d}{dx} (\frac{dy}{dx})=\frac{d}{dx}\) (-2)=0
advertisement

6. If y=log⁡(2x3), find \(\frac{d^2 y}{dx^2}\).
a) –\(\frac{2}{x^2}\)
b) \(\frac{3}{x^2}\)
c) \(\frac{2}{x^2}\)
d) –\(\frac{3}{x^2}\)
View Answer

Answer: d
Explanation: Given that, y=log⁡(2x3)
\(\frac{dy}{dx}=\frac{1}{(2x^3)}.6x^2=\frac{3}{x}\)
\(\frac{d^2 y}{dx^2}=-\frac{3}{x^2}\)

7. Find \(\frac{d^2 y}{dx^2}\)-6 \(\frac{dy}{dx}\) if y=4x4+2x.
a) \((4x^2+8x-1)\)
b) \(12(4x^2+8x-1)\)
c) –\(12(4x^2+8x-1)\)
d) \(12(4x^2-8x-1)\)
View Answer

Answer: d
Explanation: Given that, \(y=4x^4+2x\)
\(\frac{dy}{dx}\)=16x3+2
\(\frac{d^2 y}{dx^2}\)=48x2
\(\frac{d^2 y}{dx^2}\)-6 \(\frac{dy}{dx}=48x^2-96x^3-12\)
=12(4x2-8x-1)
advertisement

8. Find the second order derivative y=e2x+sin-1⁡ex .
a) e2x+\(\frac{e^x}{(1-e^2x)^{3/2}}\)
b) 4e2x+\(\frac{1}{(1-e^2x)^{3/2}}\)
c) 4e2x–\(\frac{e^x}{(1-e^2x)^{3/2}}\)
d) 4e2x+\(\frac{e^x}{(1-e^2x)^{3/2}}\)
View Answer

Answer: d
Explanation: Given that, y=e2x+sin-1⁡ex
\(\frac{dy}{dx}\)=2e2x+\(\frac{1}{\sqrt{1-e^{2x}}} e^x\)
\(\frac{d^2 y}{dx^2} = 4e^2x+\bigg(\frac{\frac{d}{dx} (e^x) \sqrt{1-e^{2x}} – \frac{d}{dx} (\sqrt{1-e^{2x}}).e^x}{(\sqrt{1-e^{2x}})^2}\bigg)\)
\(=4e^{2x}+\frac{(e^x \sqrt{1-e^{2x}})-e^x \left(\frac{1}{2\sqrt{1-e^{2x}}}.-2e^{2x}\right)}{1-e^{2x}}\)
\(=4e^{2x}+\frac{(e^x (1-e^{2x})+e^{3x})}{(1-e^{2x})^{\frac{3}{2}}}\)
\(=4e^{2x}+\frac{e^x (1-e^{2x}+e^{2x})}{(1-e^{2x})^{\frac{3}{2}}}\)
4e2x+\(\frac{e^x}{(1-e^2x)^{3/2}}\).

9. Find the second order derivative of y=3x2 1 + log⁡(4x)
a) 3+\(\frac{1}{x^2}\)
b) 3-\(\frac{1}{x^2}\)
c) 6-\(\frac{1}{x^2}\)
d) 6+\(\frac{1}{x^2}\)
View Answer

Answer: c
Explanation: Given that, y=3x2+log⁡(4x)
\(\frac{dy}{dx}=6x+\frac{1}{4x}.4=6x+\frac{1}{x}=\frac{6x^2+1}{x}\)
\(\frac{d^2 y}{dx^2}=\frac{\frac{d}{dx} (6x^2+1).(x)-\frac{d}{dx} (x).(6x^2+1)}{x^2} \Big(using\, \frac{d}{dx} (\frac{u}{v})=\frac{(\frac{d}{dx} (u).v-\frac{d}{dx} (v).u)}{v^2}\Big)\)
\(\frac{d^2 y}{dx^2}=\frac{(12x.x-6x^2-1)}{x^2} \)
\(\frac{d^2 y}{dx^2}=\frac{6x^2-1}{x^2} = 6-\frac{1}{x^2}\).
advertisement

10. Find the second order derivative if y=e2x2.
a) 4e2x2 (4x2+3)
b) 4e2x2 (4x2-1)
c) 4e2x2 (4x2+1)
d) e2x2 (4x2+1)
View Answer

Answer: c
Explanation: Given that, y=e2x2
\(\frac{dy}{dx}\)=e2x2.4x
By using u.v rule, we get
\(\frac{d^2 y}{dx^2}=\frac{d}{dx} (e^{{2x}^2}).4x+\frac{d}{dx} (4x).e^{{2x}^2}\)
16x2 e2x2+4e2x2=4e2x2 (4x2+1)

Sanfoundry Global Education & Learning Series – Mathematics – Class 12.

To practice all areas of Mathematics, here is complete set of 1000+ Multiple Choice Questions and Answers.

Participate in the Sanfoundry Certification contest to get free Certificate of Merit. Join our social networks below and stay updated with latest contests, videos, internships and jobs!

advertisement
advertisement
Manish Bhojasia - Founder & CTO at Sanfoundry
Manish Bhojasia, a technology veteran with 20+ years @ Cisco & Wipro, is Founder and CTO at Sanfoundry. He is Linux Kernel Developer & SAN Architect and is passionate about competency developments in these areas. He lives in Bangalore and delivers focused training sessions to IT professionals in Linux Kernel, Linux Debugging, Linux Device Drivers, Linux Networking, Linux Storage, Advanced C Programming, SAN Storage Technologies, SCSI Internals & Storage Protocols such as iSCSI & Fiber Channel. Stay connected with him @ LinkedIn | Youtube | Instagram | Facebook | Twitter