In the following find the approximate values, using differentials :
`(401)^(1//2)`

Answer

To find the approximate value of \( (401)^{1/2} \) using differentials, we can follow these steps:

### Step 1: Define the function
Let \( f(x) = \sqrt{x} \).

### Step 2: Choose a point close to 401
We can choose \( x = 400 \) because it is a perfect square and close to 401.

### Step 3: Calculate \( \delta x \)
Since we are approximating \( \sqrt{401} \), we have:
\[ \delta x = 401 - 400 = 1 \]

### Step 4: Find the derivative \( f'(x) \)
The derivative of \( f(x) = \sqrt{x} \) is:
\[ f'(x) = \frac{1}{2\sqrt{x}} \]

### Step 5: Evaluate the derivative at \( x = 400 \)
Now, we need to evaluate the derivative at \( x = 400 \):
\[ f'(400) = \frac{1}{2\sqrt{400}} = \frac{1}{2 \times 20} = \frac{1}{40} \]

### Step 6: Calculate \( dy \)
Using the formula for differentials, we have:
\[ dy = f'(x) \cdot \delta x \]
Substituting the values we found:
\[ dy = \frac{1}{40} \cdot 1 = \frac{1}{40} \]

### Step 7: Approximate \( f(401) \)
Now, we can find the approximate value of \( \sqrt{401} \):
\[ f(401) \approx f(400) + dy \]
Since \( f(400) = \sqrt{400} = 20 \):
\[ f(401) \approx 20 + \frac{1}{40} \]

### Step 8: Convert \( \frac{1}{40} \) to decimal
Calculating \( \frac{1}{40} \):
\[ \frac{1}{40} = 0.025 \]

### Step 9: Final approximation
Thus, we have:
\[ f(401) \approx 20 + 0.025 = 20.025 \]

### Conclusion
The approximate value of \( \sqrt{401} \) is \( \boxed{20.025} \).

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