2021 ASRJC Promo Q10

A particle is moving on the curve with equation

\(y={{\sin }^{-1}}\left( 3x \right)\), \(-\frac{1}{3}\le x\le \frac{1}{3}\),

where \(\left( x,y \right)\) is the coordinate of the particle at time \(t\) relative to a fixed point \(O\). The \(x\) and \(y\) values represent the horizontal and vertical displacements.

When the \(y\)- coordinate of the particle is \(\frac{\pi }{3}\), the rate at which the \(y\)- coordinate is decreasing with time \(t\) is \(2\) units per second. At this instant, find the exact rate at which the \(x\)-coordinate of the particle changes with time.[4]