Let $A$ be the area of a circle with radius $r$. If $\frac{d r}{d t}=4$, find $\frac{d A}{d t}$ when $r=1$.

Step 1 ：First, we know that the area of a circle is given by the formula \(A = \pi r^2\).

Step 2 ：Then, we can differentiate both sides of the equation with respect to \(t\) to get \(\frac{d A}{d t} = 2\pi r \frac{d r}{d t}\).

Step 3 ：Given that \(\frac{d r}{d t} = 4\), we can substitute this into the equation to get \(\frac{d A}{d t} = 2\pi r * 4\).

Step 4 ：Finally, we substitute \(r = 1\) into the equation to get \(\frac{d A}{d t} = 2\pi * 1 * 4\).

Step 5 ：So, \(\frac{d A}{d t} = \boxed{8\pi}\) when \(r = 1\).