Mathematics Questions and Answers – Second Order Derivatives

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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
\(\frac{dy}{dx}=9.\frac{1}{t^3}.3t^2=\frac{27}{t}\)
\(\frac{d^2 y}{dx^2}=27(-\frac{1}{t^2})=-\frac{27}{t^2}\).
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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
∴\(\left (\frac{dy}{dx}\right )^2≠\frac{d^2 y}{dx^2}\)
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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
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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)
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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)
\(\frac{dy}{dx}=6x+\frac{1}{4x}.4=6x+\frac{1}{x}=\frac{6x^2+1}{x}\)
\(\frac{d^2 y}{dx^2}=\frac{\frac{d}{dx} (6x^2+1).(x)-\frac{d}{dx} (x).(6x^2+1)}{x^2} \Big(using\, \frac{d}{dx} (\frac{u}{v})=\frac{(\frac{d}{dx} (u).v-\frac{d}{dx} (v).u)}{v^2}\Big)\)
\(\frac{d^2 y}{dx^2}=\frac{(12x.x-6x^2-1)}{x^2} \)
\(\frac{d^2 y}{dx^2}=\frac{6x^2-1}{x^2} = 6-\frac{1}{x^2}\).
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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.

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Manish Bhojasia, a technology veteran with 20+ years @ Cisco & Wipro, is Founder and CTO at Sanfoundry. He lives in Bangalore, and focuses on development of Linux Kernel, SAN Technologies, Advanced C, Data Structures & Alogrithms. Stay connected with him at LinkedIn.

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