It is given that \(y=\frac{\ln \left(3 x^{2}-1\right)}{x+2}\), for \(x\gt \frac{1}{\sqrt{3}}\). When \(x=1, y\) is increasing at the rate of \(h\) units per second. Find, in terms of \(h\), the corresponding rate of change in \(x\), giving your answer in exact form.