Question

If y = e^x cos x, prove that $\frac{d^{2}y}{d{x}^2}=2e^x\cos \left(x+\frac{\mathrm{\pi }}{2}\right)$.

Answer 1

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

Hence proved.