`sqrt(25.3)`

Answer

To evaluate \( \sqrt{25.3} \) using the method of derivatives, we can follow these steps:

### Step 1: Rewrite the expression
We can express \( 25.3 \) as \( 25 + 0.3 \). Thus, we want to evaluate \( \sqrt{25 + 0.3} \).

### Step 2: Define the function
Let \( f(x) = \sqrt{x} \). We will evaluate \( f(25.3) \) using the linear approximation.

### Step 3: Find the derivative
The derivative of \( f(x) \) is:
\[ f'(x) = \frac{1}{2\sqrt{x}} \]
Now we need to evaluate this at \( x = 25 \).

### Step 4: Calculate the derivative at \( x = 25 \)
Substituting \( x = 25 \) into the derivative:
\[ f'(25) = \frac{1}{2\sqrt{25}} = \frac{1}{2 \times 5} = \frac{1}{10} \]

### Step 5: Use linear approximation
Using the linear approximation formula:
\[ f(x + \Delta x) \approx f(x) + f'(x) \Delta x \]
where \( \Delta x = 0.3 \) (since we are moving from \( 25 \) to \( 25.3 \)).

### Step 6: Calculate \( f(25) \)
Now we calculate \( f(25) \):
\[ f(25) = \sqrt{25} = 5 \]

### Step 7: Substitute values into the linear approximation
Now we substitute the values into the linear approximation:
\[ f(25.3) \approx f(25) + f'(25) \cdot 0.3 \]
\[ f(25.3) \approx 5 + \left(\frac{1}{10}\right) \cdot 0.3 \]
\[ f(25.3) \approx 5 + 0.03 = 5.03 \]

### Conclusion
Thus, the approximate value of \( \sqrt{25.3} \) is \( 5.03 \).

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