A IIT-2007(3-marks) Maths question Que. $$ {\frac{d^{2}x}{dy^{2}}} $$ equals
(a) $$ ({\frac{d^{2}y}{dx^{2}}})^{-1}$$ (b)$$ -({\frac{d^{2}y}{dx^{2}}})^{-1}({\frac{dy}{dx}})^{-3} $$ (c)$$({\frac{d^{2}y}{dx^{2}}})({\frac{dy}{dx}})^{-2}$$ (d) $$ -({\frac{d^{2}y}{dx^{2}}})({\frac{dy}{dx}})^{-3}$$ Soln. $$ {\frac {d^{2}x}{dy^2}}$$ this can be written as $$ {\frac {d}{dy}} ({\frac {dx}{dy}}) $$ rearranging terms $$\to {\frac {d}{dx}}({\frac{1}{\frac{dy}{dx}}})(\frac{dx}{dy})$$ multiplying and dividing by dx $$ \to - (\frac{1}{\frac{dy}{dx}})({\frac{dy}{dx}})^{-2} ({\frac{d^{2}y}{dx^{2}}}) $$ differentiating ,and rearranging terms$$\to -({\frac{d^{2}y}{dx^{2}}})({\frac{dy}{dx}})^{-3} $$
Amazing isn't it!!!

a IIT -2007 (3-marks ) Maths question