A) \[-\left( \frac{{{d}^{2}}y}{d{{x}^{2}}} \right){{\left( \frac{dy}{dx} \right)}^{-3}}\]
B) \[{{\left( \frac{{{d}^{2}}y}{d{{x}^{2}}} \right)}^{-1}}\]
C) \[-{{\left( \frac{{{d}^{2}}y}{d{{x}^{2}}} \right)}^{-1}}{{\left( \frac{dy}{dx} \right)}^{-3}}\]
D) \[\left( \frac{{{d}^{2}}y}{d{{x}^{2}}} \right){{\left( \frac{dy}{dx} \right)}^{-2}}\]
Correct Answer: A
Solution :
We have \[\frac{{{d}^{2}}x}{d{{y}^{2}}}=\frac{d}{dy}\left( \frac{dx}{dy} \right)=\frac{d}{dy}\left( \frac{1}{\frac{dy}{dx}} \right)\] \[=\frac{d}{dx}\left( \frac{1}{\frac{dy}{dx}} \right).\frac{dx}{dy}=-\frac{1}{{{\left( \frac{dy}{dx} \right)}^{2}}}.\frac{{{d}^{2}}y}{d{{x}^{2}}}.\frac{1}{\left( \frac{dy}{dx} \right)}\] \[=-\frac{1}{{{\left( \frac{dy}{dx} \right)}^{3}}}.\frac{{{d}^{2}}y}{d{{x}^{2}}}=-{{\left( \frac{dy}{dx} \right)}^{-3}}\left( \frac{{{d}^{2}}y}{d{{x}^{2}}} \right)\]
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