If \( x^{2}y^{2}=a^{\pi/2} \), then \( \displaystyle\frac{d^{2}y}{dx^{2}}= \)
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \( \displaystyle\frac{2y^{3}}{a^{\pi/2}} \)
The right side \( a^{\pi/2} \) is a constant, so the relation says that \( xy \) is constant.
Write \( a^{\pi/2}=k^{2} \). Then \( xy=k \), that is
\( \displaystyle y=\frac{k}{x} \)
Differentiating twice,
\( \displaystyle \frac{dy}{dx}=-\frac{k}{x^{2}} \)
\( \displaystyle \frac{d^{2}y}{dx^{2}}=\frac{2k}{x^{3}} \)
The options give the result in terms of y. Since \( y^{3}=\frac{k^{3}}{x^{3}} \),
\( \displaystyle \frac{2k}{x^{3}}=\frac{2}{k^{2}}\cdot\frac{k^{3}}{x^{3}} \)
\( \displaystyle =\frac{2y^{3}}{k^{2}} \)
\( \displaystyle =\frac{2y^{3}}{a^{\pi/2}} \)