Let $x y=4$ and $\frac{d y}{d t}=3$ Find $\frac{d x}{d t}$ when $x=4$

Step 1 ：Given the equation \(xy = 4\), we can differentiate both sides with respect to \(t\) to get \(x\frac{dy}{dt} + y\frac{dx}{dt} = 0\).

Step 2 ：We are given that \(\frac{dy}{dt} = 3\) and we need to find \(\frac{dx}{dt}\) when \(x = 4\).

Step 3 ：We can substitute \(x = 4\) into the original equation to find the corresponding value of \(y\). So, \(y = \frac{4}{x} = 1\).

Step 4 ：Then we can substitute these values into the differentiated equation to solve for \(\frac{dx}{dt}\). So, \(4*3 + 1*\frac{dx}{dt} = 0\). Solving this equation gives \(\frac{dx}{dt} = -12\).

Step 5 ：Final Answer: The rate of change of \(x\) with respect to \(t\) when \(x = 4\) is \(\boxed{-12}\).