Answer: The approximate change in \(y\) is \(0.651h\).

Answer: The approximate change in \(y\) is \(0.651h\).

Write

\(y=(1+\sin 3x)^4x^{-1/2}\).

Differentiate using the product rule. Let \(u=(1+\sin 3x)^4\) and \(v=x^{-1/2}\).

Then

\(\displaystyle \frac{du}{dx}=4(1+\sin 3x)^3(3\cos 3x)=12(1+\sin 3x)^3\cos 3x\),

and

\(\displaystyle \frac{dv}{dx}=-\frac{1}{2}x^{-3/2}\).

So

\(\displaystyle \frac{dy}{dx}=12(1+\sin 3x)^3\cos 3x\,x^{-1/2}-\frac{1}{2}(1+\sin 3x)^4x^{-3/2}\).

At \(x=1.9\), this gives

\(\displaystyle \frac{dy}{dx}=0.651\ldots\).

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

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

Therefore the approximate change is

\(0.651h\).