Q104 / 160 · AP EAMCET 2022 (05 Jul Shift 1) · Mathematics

If $x=f(\theta)$ and $y=g(\theta)$, then $\frac{d^2 y}{d x^2}=$

  • A
    $\frac{g^{\prime \prime}(\theta)}{f^{\prime}(\theta)}$
  • B
    $\frac{f^{\prime \prime}(\theta)}{x(\theta)}$
  • Correct answer
    C
    $\frac{f^{\prime}(\theta) g^{\prime \prime}(\theta)-g^{\prime}(\theta) f^{\prime \prime}(\theta)}{\left(f^{\prime}(\theta)\right)^3}$
  • D
    $\frac{g^{\prime}(\theta) f^{\prime \prime}(\theta)-g^{\prime \prime}(\theta) f^{\prime \prime}(\theta)}{\left(g^{\prime \prime}(\theta)\right)^3}$

Solution

Given, $x=f(\theta)$ and $y=g(\theta)$ $\frac{d x}{d \theta}=f^{\prime}(\theta), \frac{d y}{d \theta}=g^{\prime}(\theta)$ $\Rightarrow \quad \frac{d y}{d x}=\frac{g^{\prime}(\theta)}{f^{\prime}(\theta)}$ $\begin{aligned} & \Rightarrow \frac{d^2 y}{d x^2}=\left(\frac{g^{\prime \prime}(\theta) f^{\prime}(\theta)-f^{\prime \prime}(\theta) g^{\prime}(\theta)}{\left(f^{\prime}(\theta)^2\right)}\right) \frac{d \theta}{d x} \\ & \Rightarrow \frac{d^2 y}{d x^2}=\left(\frac{g^{\prime \prime}(\theta) f^{\prime}(\theta)-f^{\prime \prime}(\theta) g^{\prime}(\theta)}{\left(f^{\prime}(\theta)\right)^3}\right)\end{aligned}$

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

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