Compute the derivative without using a calculator. (Your answer should be exact.) \( f(x)=\tan ^{-1}(x), x=0.25 \) \( f^{\prime}(0.25)=\square \)

To find the derivative of \( f(x) = \tan^{-1}(x) \), we use the formula for the derivative of the arctangent function: \[ f'(x) = \frac{1}{1 + x^2} \] Now, we need to evaluate this derivative at \( x = 0.25 \): \[ f'(0.25) = \frac{1}{1 + (0.25)^2} \] Calculating \( (0.25)^2 = 0.0625 \), we get: \[ f'(0.25) = \frac{1}{1 + 0.0625} = \frac{1}{1.0625} \] To express \( 1.0625 \) as a fraction: \[ 1.0625 = \frac{17}{16} \implies f'(0.25) = \frac{1}{\frac{17}{16}} = \frac{16}{17} \] Thus, \[ f'(0.25) = \frac{16}{17} \] So, the derivative evaluated at \( x = 0.25 \) is \( \frac{16}{17} \).