Final answer:
By differentiating the equation \(xy=4\) with respect to time, we can solve for \(\frac{dy}{dt}\) and \(\frac{dx}{dt}\) given certain values of \(x\), \(y\), and one of the derivatives. We find \(\frac{dy}{dt} = -13\) when \(x = 4\) and \(\frac{dx}{dt} = 13\), and \(\frac{dx}{dt}=\frac{9}{4}\) when \(x=1\) and \(\frac{dy}{dt}=-9\).
Step-by-step explanation:
To find \(\frac{dy}{dt}\) when \(x=4\), we use the given \(xy=4\). First, differentiate both sides with respect to \(t\) to get \(x\frac{dy}{dt} + y\frac{dx}{dt}=0\). Plug in \(x=4\) and \(\frac{dx}{dt}=13\), we get \(4\frac{dy}{dt} + 4 \cdot 13 = 0\). Therefore, \(\frac{dy}{dt} = -13\).
For the second part, we again differentiate \(xy=4\), this time plugging in \(x=1\) and \(\frac{dy}{dt}=-9\), we find that \(1\cdot(-9) + y\frac{dx}{dt}=0\). Since \(xy=4\), when \(x=1\), \(y=4\), which gives us \(-9 + 4\frac{dx}{dt}=0\) so that \(\frac{dx}{dt}=\frac{9}{4}\).