NIMCET 2018 — Mathematics PYQ

Concept:

$$\overline{\frac{{\mathrm{d}}^2\mathrm{x}}{\mathrm{d}{\mathrm{y}}^2}=\frac{\mathrm{d}\left(\frac{\mathrm{dx}}{\mathrm{dy}}\right)}{\mathrm{dy}}}$$

testlo

$$\frac{\mathrm{d}}{\mathrm{dx}}(\frac{1}{\mathrm{y}})=-\frac{1}{{\mathrm{y}}^2}\frac{\mathrm{dy}}{\mathrm{dx}}$$

Calculation:

\\ \frac{\mathrm{d}^{2} \mathrm{x}}{\mathrm{d} y^{2}}=\frac{\mathrm{d}\left(\frac{\mathrm{d} x}{\mathrm{~d} y}\right)}{\mathrm{d} y} \\ =\mathrm{d}\left(\frac{1}{\frac{\mathrm{d} y}{\mathrm{~d} x}}\right) \\ =\frac{\mathrm{d} y}{\mathrm{~d} x} \times \frac{-1}{\left(\frac{\mathrm{d} y}{\mathrm{~d} x}\right)^{2}} \times \frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}} \\ =-\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}\left(\frac{\mathrm{d} y}{\mathrm{~d} x}\right)^{-3} \\ \text { Hence, option (4) is correct. } \end{array}