Given the curve \(y = 18 - \frac{3}{8}x^{\frac{5}{2}}\), we need to find \(\frac{dy}{dt}\) when \(x = 4\) and \(\frac{dx}{dt} = 2\).

First, find \(\frac{dy}{dx}\):

\(\frac{dy}{dx} = -\frac{5}{2} \times \frac{3}{8} x^{\frac{3}{2}} = -\frac{15}{16} x^{\frac{3}{2}}\).

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

Substitute \(x = 4\) and \(\frac{dx}{dt} = 2\):

\(\frac{dy}{dt} = -\frac{15}{16} \times 4^{\frac{3}{2}} \times 2\).

Calculate \(4^{\frac{3}{2}} = 8\), so:

\(\frac{dy}{dt} = -\frac{15}{16} \times 8 \times 2 = -15\).