Question

Find the second order derivative of the given functions.

$e^{x}sin5x$

Answer 1

Ley y = $e^{x}sin5x$. Differentiating twicely w.r.t. x, we get

$\frac{dy}{dx}=\frac{d}{dx}\left(e^{x}sin5x\right)=e^x\frac{d}{dx}\left(sin5x\right)+sin5x\frac{d}{dx}\left(e^x\right)=e^{x}cos5x.5+sin5x{e}^x\left(Usingproductrule\right)=e^x(5cos5x+sin5x)and\frac{d^{2}y}{d{x}^2}=\frac{d}{dx}\left\{e^x\right(5cos5x+sin5x\left)\right\}=e^x\frac{d}{dx}(5cos5x+sin5x)+(5cos5x+sin5x)\frac{d}{dx}{e}^x=e^x\{-5sin5x.5+cos5x.5\}+(5cos5x+sin5x)e^x=e^x\{10cos5x-24sin5x\}=2e^x(5cos5x-12sin5x)$