Question

Find the second order derivatives of the function
$e^xsin5x$

Answer 1

The correct answer is $2e^x(5cos5x-12sin5x)$
Let $y=e^xsin5x$
Then,
$\frac{dy}{dx}=\frac{d}{dx}(e^xsin5x)=sin5x.\frac{d}{dx}(e^x)+e^x\frac{d}{dx}(sin5x)$
$=sin5x.e^x+e^x.cos5x\frac{d}{dx}(5x)=e^xsin5x+e^xcos5x.5$
$=e^x(sin5x+5cos5x)$
$∴\frac{d^2y}{dx^2}=\frac{d}{dx}[e^x(sin5x+5cos5x)]$
$=(sin5x+5cos5x).\frac{d}{dx}(e^x)+e^x.\frac{d}{dx}(sin5x+5cos5x)$
$=(sin5x+5cos5x)e^x+e^x[cos5x.\frac{d}{dx}(5x)+5(-sin5x).\frac{d}{dx}(5x)]$
$=e^x(sin5x+5cos5x)+e^x(5cos5x-25sin5x)$
Then, $e^x(10cos5x-24sin5x)=2e^x(5cos5x-12sin5x)$