Question

If x = t² and y = t³, find$\frac{d^{2}y}{d{x}^2}$.

Answer 1

Here,
$$\begin{gathered} x=t^2\mathrm{and}y=t^3 \\ \Rightarrow \frac{\mathit{d}\mathit{x}}{\mathit{d}\mathit{t}}=2t\mathrm{and}\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{t}}=3t^2 \\ \therefore \frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}=\frac{3t}{2} \\ \Rightarrow \frac{{\mathit{d}}^{\mathit{2}}y}{\mathit{d}{x}^{\mathit{2}}}=\frac{3}{2}\frac{dt}{dx}=\frac{3}{4t} \end{gathered}$$