Q63 / 125 · BITSAT 2016 · Mathematics
For any differentiable function $y$ of $x$, $\frac{d^{2} x}{d y^{2}}\left(\frac{d y}{d x}\right)^{3}+\frac{d^{2} y}{d x^{2}}=$
- Correct answerA0
- B$y$
- C$-y$
- D$x$
Solution
$\frac{d y}{d x}=\left(\frac{d x}{d y}\right)^{-1}$
$\Rightarrow \frac{d^{2} y}{d x^{2}}=-1\left(\frac{d x}{d y}\right)^{-2}\left\{\frac{d}{d x}\left(\frac{d x}{d y}\right)\right\}$
$\Rightarrow \frac{d^{2} y}{d x^{2}}=(-1)\left(\frac{d x}{d y}\right)^{-2}\left\{\frac{d}{d y}\left(\frac{d x}{d y}\right) \frac{d y}{d x}\right\}$
$=(-1)\left(\frac{d y}{d x}\right)^{2}\left\{\frac{d^{2} x}{d y^{2}} \cdot \frac{d y}{d x}\right\}=-\left(\frac{d y}{d x}\right)^{3}\left\{\frac{d^{2} x}{d y^{2}}\right\}$
$\Rightarrow \frac{d^{2} x}{d y^{2}}\left(\frac{d y}{d x}\right)^{3}+\frac{d^{2} y}{d x^{2}}=0$
Hence, the correct answer is option: 'A'.