Answer: Approximately \(7.14h\).

Write

\(y=e^{3x}\sin x\,x^{-2}.\)

Using the product rule,

\(\frac{dy}{dx}=e^{3x}x^{-2}(3\sin x+\cos x)-2e^{3x}\sin x\,x^{-3}.\)

Factorising gives

\(\frac{dy}{dx}=\frac{e^{3x}}{x^3}\{x(3\sin x+\cos x)-2\sin x\}.\)

For a small increase \(h\), the approximate change in \(y\) is

\(\Delta y\approx h\frac{dy}{dx}.\)

At \(x=0.5\),

\(\frac{dy}{dx}\approx 7.13766.\)

Therefore the approximate change in \(y\) is

\(\boxed{7.14h}.\)