`sqrt(401)`

Answer

To evaluate \(\sqrt{401}\) using differentials, we can follow these steps:

### Step 1: Identify a nearby perfect square
We can express \(401\) as \(400 + 1\). Here, we let \(x = 400\) and \(\Delta x = 1\).

### Step 2: Define the function
Let \(f(x) = \sqrt{x} = x^{1/2}\).

### Step 3: Calculate \(f(x)\)
Now, we need to calculate \(f(x)\) at \(x = 400\):
\[ f(400) = \sqrt{400} = 20. \]

### Step 4: Find the derivative \(f'(x)\)
Next, we find the derivative of \(f(x)\):
\[ f'(x) = \frac{1}{2} x^{-1/2} = \frac{1}{2\sqrt{x}}. \]
Now, we can evaluate the derivative at \(x = 400\):
\[ f'(400) = \frac{1}{2\sqrt{400}} = \frac{1}{2 \cdot 20} = \frac{1}{40}. \]

### Step 5: Use the differential approximation
Using the differential approximation:
\[ f(x + \Delta x) \approx f(x) + f'(x) \cdot \Delta x. \]
Substituting the values we have:
\[ f(401) \approx f(400) + f'(400) \cdot 1. \]
This gives us:
\[ f(401) \approx 20 + \frac{1}{40} \cdot 1 = 20 + 0.025 = 20.025. \]

### Conclusion
Thus, the approximate value of \(\sqrt{401}\) is \(20.025\).

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