Question

Find the second order derivative y=e^2x+sin^-1⁡e^x .

(a) e^2x+\(\frac{e^x}{(1-e^2x)^{3/2}}\)

(b) 4e^2x+\(\frac{1}{(1-e^2x)^{3/2}}\)

(c) 4e^2x–\(\frac{e^x}{(1-e^2x)^{3/2}}\)

(d) 4e^2x+\(\frac{e^x}{(1-e^2x)^{3/2}}\)

I got this question in an internship interview.

My question is based upon Second Order Derivatives topic in section Continuity and Differentiability of Mathematics – Class 12

Answer 1

Correct answer is (d) 4e^2x+\(\frac{e^x}{(1-e^2x)^{3/2}}\)

The explanation: Given that, y=e^2x+sin^-1⁡e^x

\(\frac{dy}{dx}\)=2e^2x+\(\frac{1}{\sqrt{1-e^{2x}}} e^x\)

\(\frac{d^2 y}{dx^2} = 4e^2x+\bigg(\frac{\frac{d}{dx} (e^x) \sqrt{1-e^{2x}} – \frac{d}{dx} (\sqrt{1-e^{2x}}).e^x}{(\sqrt{1-e^{2x}})^2}\bigg)\)

\(=4e^{2x}+\frac{(e^x \sqrt{1-e^{2x}})-e^x \left(\frac{1}{2\sqrt{1-e^{2x}}}.-2e^{2x}\right)}{1-e^{2x}}\)

\(=4e^{2x}+\frac{(e^x (1-e^{2x})+e^{3x})}{(1-e^{2x})^{\frac{3}{2}}}\)

\(=4e^{2x}+\frac{e^x (1-e^{2x}+e^{2x})}{(1-e^{2x})^{\frac{3}{2}}}\)

4e^2x+\(\frac{e^x}{(1-e^2x)^{3/2}}\).