JEE Advanced · Mathematics · 24. Differentiation
\(\frac{d^2 x}{d y^2}\) equals
- A
\(\left(\frac{d^2 y}{d x^2}\right)^{-1}\)
- B
\(-\left(\frac{d^2 y}{d x^2}\right)^{-1}\left(\frac{d y}{d x}\right)^{-3}\)
- C
\(\left(\frac{d^2 y}{d x^2}\right)\left(\frac{d y}{d x}\right)^{-2}\)
- D
\(-\left(\frac{d^2 y}{d x^2}\right)\left(\frac{d y}{d x}\right)^{-3}\)
Answer & Solution
Correct Answer
(D)
\(-\left(\frac{d^2 y}{d x^2}\right)\left(\frac{d y}{d x}\right)^{-3}\)
Step-by-step Solution
Detailed explanation
Since,
\[
\begin{aligned}
\Rightarrow \quad \frac{d}{d y}\left(\frac{d x}{d y}\right) & =\frac{d}{d x}\left(\frac{d y}{d x}\right)^{-1} \frac{d x}{d y} \\
\frac{d^2 x}{d y^2} & =-\left(\frac{d^2 y}{d x^2}\right)\left(\frac{d y}{d x}\right)^{-2}\left(\frac{d x}{d y}\right) \\
& =-\left(\frac{d^2 y}{d x^2}\right)\left(\frac{d y}{d x}\right)^{-3}
\end{aligned}
\]
\[
\begin{aligned}
\Rightarrow \quad \frac{d}{d y}\left(\frac{d x}{d y}\right) & =\frac{d}{d x}\left(\frac{d y}{d x}\right)^{-1} \frac{d x}{d y} \\
\frac{d^2 x}{d y^2} & =-\left(\frac{d^2 y}{d x^2}\right)\left(\frac{d y}{d x}\right)^{-2}\left(\frac{d x}{d y}\right) \\
& =-\left(\frac{d^2 y}{d x^2}\right)\left(\frac{d y}{d x}\right)^{-3}
\end{aligned}
\]
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