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

Question

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

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Answer

Given that, $x=t^2$ and $y=t^3$
Then, $\frac{d x}{d t}=2 t$
and $\frac{d y}{d t}=3 t^2$
Thus, $\frac{d y}{d x}=\frac{3 t^2}{2 t}=\frac{3 t}{2}$
$\Rightarrow \frac{d^2 y}{d x^2}=\frac{3}{2} \frac{d t}{d x}$
$\frac{3}{2} \cdot \frac{1}{2 t}=\frac{3}{4 t}$

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