Find the approximate value of `tan^(-1)(0.999)` using differential.

Answer

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

### Step-by-Step Solution:

1. **Define the function**: Let \( f(x) = \tan^{-1}(x) \).

2. **Find the derivative**: The derivative of \( f(x) \) is given by:
\[ f'(x) = \frac{1}{1 + x^2} \]

3. **Choose a point for approximation**: We want to approximate \( \tan^{-1}(0.999) \). We can choose \( a = 1 \) because \( 0.999 \) is close to \( 1 \).

4. **Calculate \( h \)**: Since \( 0.999 = 1 - 0.001 \), we have:
\[ h = -0.001 \]

5. **Use the linear approximation formula**: The linear approximation formula is:
\[ f(a + h) \approx f(a) + f'(a) \cdot h \]

6. **Calculate \( f(a) \)**: We know:
\[ f(1) = \tan^{-1}(1) = \frac{\pi}{4} \]

7. **Calculate \( f'(a) \)**: Now, we need to find \( f'(1) \):
\[ f'(1) = \frac{1}{1 + 1^2} = \frac{1}{2} \]

8. **Substitute values into the approximation formula**:
\[ f(0.999) \approx f(1) + f'(1) \cdot h \]
Substituting the values we calculated:
\[ f(0.999) \approx \frac{\pi}{4} + \frac{1}{2} \cdot (-0.001) \]
\[ f(0.999) \approx \frac{\pi}{4} - 0.0005 \]

9. **Final approximation**: Thus, the approximate value of \( \tan^{-1}(0.999) \) is:
\[ \tan^{-1}(0.999) \approx \frac{\pi}{4} - 0.0005 \]