Step 1 :Given the equation \(x^{2}+y^{2}=9\), we can differentiate both sides with respect to \(t\) to get \(2x\frac{dx}{dt} + 2y\frac{dy}{dt} = 0\).
Step 2 :We can simplify this to \(x\frac{dx}{dt} + y\frac{dy}{dt} = 0\).
Step 3 :Rearranging for \(\frac{dy}{dt}\), we get \(\frac{dy}{dt} = -\frac{x}{y}\frac{dx}{dt}\).
Step 4 :Substituting the given values \(x=2\), \(y=\sqrt{5}\), and \(\frac{dx}{dt}=-1\) into the equation, we get \(\frac{dy}{dt} = -\frac{2}{\sqrt{5}}(-1)\).
Step 5 :Simplifying this, we get \(\frac{dy}{dt} = \boxed{\sqrt{\frac{2}{\sqrt{5}}}}\).