In the following find the approximate values, using differentials :
`(0.999)^(1//10)`

Answer

To find the approximate value of \( (0.999)^{\frac{1}{10}} \) using differentials, we can follow these steps:

### Step 1: Define the function
Let \( y = x^{\frac{1}{10}} \).

### Step 2: Choose a point near the value of interest
We want to evaluate \( y \) at \( x = 0.999 \). We can choose \( x = 1 \) as a point close to \( 0.999 \). Thus, we have:
- \( x = 1 \)
- \( \Delta x = 0.999 - 1 = -0.001 \)

### Step 3: Calculate the derivative
Now we need to find the derivative \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{1}{10} x^{-\frac{9}{10}} \]

### Step 4: Evaluate the derivative at \( x = 1 \)
Substituting \( x = 1 \) into the derivative:
\[ \frac{dy}{dx} \bigg|_{x=1} = \frac{1}{10} \cdot 1^{-\frac{9}{10}} = \frac{1}{10} \]

### Step 5: Use the differential to approximate \( \Delta y \)
The differential \( dy \) can be approximated as:
\[ dy \approx \frac{dy}{dx} \bigg|_{x=1} \cdot \Delta x = \frac{1}{10} \cdot (-0.001) = -0.0001 \]

### Step 6: Calculate the approximate value of \( y \)
Now, we can find the approximate value of \( y \) at \( x = 0.999 \):
\[ y \approx y(1) + dy = 1 + (-0.0001) = 1 - 0.0001 = 0.9999 \]

Thus, the approximate value of \( (0.999)^{\frac{1}{10}} \) is:
\[ \boxed{0.9999} \]