Use the differential to approximate the radical expression. Then use a calculator to approximate the quantity, and give the absolute value of the difference in the two results to 4 decimal places.
\[
\sqrt{83}
\]
What is the value found using the differential?
$\sqrt{83} \approx$
(Round to four decimal places as needed.)

Step 1 ：Choose a number close to 83 that we can easily take the square root of. A good choice is 81 because its square root is 9.

Step 2 ：Use the function \(f(x) = \sqrt{x}\) and find its derivative \(f'(x) = \frac{1}{2\sqrt{x}}\).

Step 3 ：The differential of f at x = 81 is \(df = f'(81)dx = \frac{1}{2\sqrt{81}}dx = \frac{1}{2*9}dx = \frac{1}{18} dx\).

Step 4 ：We want to approximate \(\sqrt{83}\), so we choose \(dx = 83 - 81 = 2\).

Step 5 ：Substituting \(dx = 2\) into the differential, we get \(df = \frac{1}{18} * 2 = \frac{1}{9}\).

Step 6 ：\(\sqrt{83}\) is approximately \(\sqrt{81} + df = 9 + \frac{1}{9} = 9.1111\) (rounded to four decimal places).

Step 7 ：So, the approximate value of \(\sqrt{83}\) is \(\boxed{9.1111}\).