Question

If y = e^−x cos x, show that $\frac{d^{2}y}{d{x}^2}=2e^{-x}\sin x$.

Answer 1

Here,

$$\begin{gathered} y=e^{-x}\cos x \\ \mathrm{Differentiating}\mathrm{w}.\mathrm{r}.\mathrm{t}.\mathit{}x,\mathrm{we}\mathrm{get} \\ \frac{\mathit{d}y}{\mathit{d}x}=-e^{-x}\sin x-e^{-x}\cos x \\ =-\left(e^{-x}\sin x+e^{-x}\cos x\right) \\ \mathrm{Differentiating}\mathrm{again}\mathrm{w}.\mathrm{r}.\mathrm{t}.\mathit{}x,\mathrm{we}\mathrm{get} \\ \frac{{\mathit{d}}^{\mathit{2}}y}{\mathit{d}{x}^2}=-\left(e^{-x}\cos x-e^{-x}\sin x-e^{-x}\sin x-e^{-x}\cos x\right) \\ =2e^{-x}\sin x \end{gathered}$$

Hence proved.