First, find \(f'(4)\) by substituting \(x = 4\) into \(f'(x) = 6x^{-\frac{1}{2}} - 4x^{-\frac{3}{2}}\).

\(f'(4) = 6(4)^{-\frac{1}{2}} - 4(4)^{-\frac{3}{2}}\)

\(= 6 \times \frac{1}{2} - 4 \times \frac{1}{8}\)

\(= 3 - 0.5\)

\(= \frac{5}{2}\)

Using the chain rule, \(\frac{dy}{dt} = \frac{dy}{dx} \times \frac{dx}{dt}\).

Given \(\frac{dx}{dt} = 0.12\), substitute \(\frac{dy}{dx} = f'(4) = \frac{5}{2}\).

\(\frac{dy}{dt} = \frac{5}{2} \times 0.12\)

\(= 0.3\)