If x=t^2, y=t^3 then find d^2 y d x^2 . - Vidyadip

Question

If $x=t^2, y=t^3$ then find $\frac{d^2 y}{d x^2}$.

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Answer

$\begin{array}{l}x=t^2 \Rightarrow \frac{d x}{d t}=2 t \\ y=t^3 \Rightarrow \frac{d y}{d t}=3 t^2\end{array}$
then\[\frac{d y}{d x}=\frac{d y / d t}{d x / d t}=\frac{3 t^2}{2 t}=\frac{3}{2} t\]
$\begin{aligned} \therefore \quad \frac{d^2 y}{d x^2} & =\frac{d}{d x}\left(\frac{3}{2} t\right)=\frac{3}{2} \cdot \frac{d t}{d x}=\frac{3}{2} \times \frac{1}{2 t} \\ & =\frac{3}{4 t}\end{aligned}$

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