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Question: Answered & Verified by Expert
If $x=a t^2, y=2 a t$, then $\frac{d^2 y}{d x^2}=$
MathematicsDifferentiationJEE Main
Options:
  • A $-\frac{1}{t^2}$
  • B $\frac{1}{2 a t^3}$
  • C $-\frac{1}{t^3}$
  • D $-\frac{1}{2 a t^3}$
Solution:
1465 Upvotes Verified Answer
The correct answer is: $-\frac{1}{2 a t^3}$
$\begin{aligned} & \frac{d y}{d x}=\frac{d y / d t}{d x / d t}=\frac{2 a}{2 a t} \Rightarrow \frac{d y}{d x}=\frac{1}{t}=\frac{2 a}{y} \\ & \Rightarrow y \frac{d y}{d x}=2 a \Rightarrow y \frac{d^2 y}{d x^2}+\left(\frac{d y}{d x}\right)^2=0 \\ & \Rightarrow \frac{d^2 y}{d x^2}=\frac{-(d y / d x)^2}{y}=\frac{-1}{2 a t^3} .\end{aligned}$

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