${e^x}\sin 5x$

Let $y = {e^x}\sin 5x$

$\Rightarrow$ $\cfrac{{dy}}{{dx}} = {e^x} \cdot \cos 5x \cdot 5 + \sin 5x \cdot {e^x}$

$= {e^x}[5\cos 5x + \sin 5x]$

therefore, $\cfrac{{{d^2}y}}{{d{x^2}}} = {e^x}[5( - \sin 5x) \cdot 5 + \cos 5x \cdot 5] + [5\cos 5x + \sin 5x]{e^x}$

$= {e^x}[ - 25\sin 5x + 5\cos 5x + 5\cos 5x + \sin 5x]$

$= {e^x}[10\cos 5x - 24\sin 5x] = 2{e^x}[5\cos 5x - 12\sin 5x]$