Question

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

Answer 1

Given,

$y=e^{-x}\cos x$

Now differentiating both sides with respect to $x$ we get,

$\frac{dy}{dx}=-e^{-x}\cdot \cos x-e^{-x}\cdot \sin x$

Again differentiating both sides with respect to $x$ we get,

$\frac{d^{2}y}{d{x}^2}=(e^{-x}\cos x+e^{-x}\sin x)+(e^{-x}\sin x-e^{-x}\cos x)$

or, $\frac{d^{2}y}{d{x}^2}=2e^{-x}\sin x$.