Question

If x = 4z² + 5, y = 6z² + 7z + 3, find $\frac{d^{2}y}{d{x}^2}$.

Answer 1

Here,
$$\begin{gathered} x=4z^2+5\mathrm{and}y=6z^2+7z+3 \\ \mathrm{Differentiating}\mathrm{w}.\mathrm{r}.\mathrm{t}.z\mathit{,}\mathit{}\mathrm{we}\mathrm{get} \\ \frac{\mathit{d}x}{\mathit{d}z}=8z\mathrm{and}\frac{\mathit{d}y}{\mathit{d}z}=12z+7 \\ \therefore \frac{\mathit{d}y}{\mathit{d}x}=\frac{12z+7}{8z} \\ \mathrm{Differentiating}\mathrm{w}.\mathrm{r}.\mathrm{t}.x\mathit{,}\mathit{}\mathrm{we}\mathrm{get} \\ \frac{{\mathit{d}}^{\mathit{2}}y}{\mathit{d}{x}^{\mathit{2}}}=\frac{12\times 8z-8\left(12z+7\right)}{64z^2}\times \frac{dz}{dx} \\ =\frac{96z-96z-56}{512z^3}=\frac{-7}{64z^3} \end{gathered}$$