Q102 / 160 · AP EAMCET 2024 (19 May Shift 2) · Mathematics

If $y=f(x)$ is a thrice differentiable function and a bijection then that $\frac{d^2 x}{d y^2}\left(\frac{d y}{d x}\right)^3+\frac{d^2 y}{d x^2}=$

  • A
    y
  • B
    -y
  • C
    x
  • Correct answer
    D
    0

Solution

Given, $y=f(x)$ is differentiable and bijective function. Now, $\frac{d x}{d y}=\frac{1}{\left(\frac{d y}{d x}\right)} \Rightarrow \frac{d}{d y}\left(\frac{d y}{d x}\right)=\frac{d}{d y}\left\{\left(\frac{d y}{d x}\right)^{-1}\right\}$ $\begin{aligned} & \Rightarrow \frac{d^2 x}{d y^2}=\frac{d}{d y}\left(\left(\frac{d y}{d x}\right)^{-1}\right) \cdot \frac{d x}{d y} \\ & \Rightarrow \frac{d^2 x}{d y^2}=-\left(\frac{d y}{d x}\right)^{-2} \cdot \frac{d^2 y}{d x^2} \times\left(\frac{d y}{d x}\right)^{-1} \\ & \Rightarrow \frac{d^2 x}{d y^2} \cdot\left(\frac{d y}{d x}\right)^3+\frac{d^2 y}{d x^2}=0\end{aligned}$

Hence, the correct answer is option: 'D'.

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