Find the differential of the function $f(x, y)=6 x \sqrt{y}$ at the point $(3,6)$ using $\Delta x=0.2$ and $\Delta y=-0.2$.
\[
d z=
\]
Now find $\Delta z$ and compare it to your answer above
\[
\Delta z=
\]

Step 1 ：First, we need to find the partial derivatives of the function \(f(x, y)=6 x \sqrt{y}\) with respect to x and y. The partial derivative with respect to x is \(\frac{\partial f}{\partial x} = 6\sqrt{y}\) and the partial derivative with respect to y is \(\frac{\partial f}{\partial y} = \frac{3x}{\sqrt{y}}\).

Step 2 ：Next, we substitute the point \((3,6)\) and the changes \(\Delta x=0.2\) and \(\Delta y=-0.2\) into the formula for the differential of a function to find \(dz\). The formula is \(dz = \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial y} dy\). Substituting the values we get \(dz = 0.9\sqrt{6}\).

Step 3 ：To find the change in z, \(\Delta z\), we calculate the difference between the value of the function at the point \((3+0.2,6-0.2)\) and the value of the function at the point \((3,6)\). This gives us \(\Delta z = 46.2397231825624 - 18\sqrt{6}\).

Step 4 ：\(\boxed{\text{Final Answer: The differential of the function } f(x, y)=6 x \sqrt{y} \text{ at the point } (3,6) \text{ using } \Delta x=0.2 \text{ and } \Delta y=-0.2 \text{ is } dz = 0.9\sqrt{6}. \text{ The change in z, } \Delta z, \text{ is } 46.2397231825624 - 18\sqrt{6}.}\)