At \(x = 1\), \(\frac{dy}{dx} = 6\).

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

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

\(\frac{dx}{dy} = \frac{1}{\frac{dy}{dx}} = \frac{1}{6}\).

Given \(\frac{dy}{dt} = 3\), substitute into the chain rule:

\(\frac{dx}{dt} = \frac{1}{6} \times 3 = \frac{1}{2}\).