Question

If x = 2at, y = at², where a is a constant, then find $\frac{d^{2}y}{d{x}^2}\mathrm{at}x=\frac{1}{2}$.

Answer 1

Here,
$$\begin{gathered} x=2at\mathrm{and}y=a{t}^2 \\ \mathrm{Differentiating}w.\mathrm{r}.\mathrm{t}.t\mathit{,}\mathit{}\mathrm{we}\mathrm{get} \\ \frac{\mathit{d}x}{\mathit{d}t}=2a\mathrm{and}\frac{\mathit{d}y}{\mathit{d}t}=2at \\ \therefore \frac{\mathit{d}y}{\mathit{d}x}=\frac{2at}{2a}=t \\ \mathrm{Differentiating}\mathrm{again}\mathrm{w}.\mathrm{r}.\mathrm{t}.t\mathit{,}\mathit{}\mathrm{we}\mathrm{get} \\ \frac{{\mathit{d}}^{\mathit{2}}y}{\mathit{d}{x}^{\mathit{2}}}=1\times \frac{dt}{dx}=\frac{1}{2a} \\ \mathrm{Now},{\left[\frac{{\mathit{d}}^{\mathit{2}}y}{\mathit{d}{x}^{\mathit{2}}}\right]}_{x=\frac{1}{2}}=\frac{1}{2a} \end{gathered}$$