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

$x=t^2$
$\Rightarrow \frac{d x}{d t}=2 t$
$y=t^3$
$\Rightarrow \frac{d y}{d t}=3 t^2$
then$\frac{d y}{d x}=\frac{d y / d t}{d x / d t}=\frac{3 t^2}{2 t}=\frac{3}{2} t$
$ \therefore \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}$

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