Question

If \[ f(t) = \frac{t}{2} - \frac{1}{4} \log(2t - 1), \] then \[ f'\left( \frac{t + 1}{2t + 1} \right) = ? \]

• A. \( t \)
• B. \( 1 + t \)
• C. \( 2t + 1 \)
• D. \( t - 1 \)

Hint:

When evaluating derivative at a transformed point, consider substitution or inverse relation.

Answer 1

The Correct Option is B

Differentiate \( f(t) \): \[ f'(t) = \frac{1}{2} - \frac{1}{4} \cdot \frac{2}{2t - 1} = \frac{1}{2} - \frac{1}{2t - 1} \] Substitute \( t = \frac{t + 1}{2t + 1} \) into derivative: Use substitution or inverse function trick. Eventually simplifies to: \[ f'\left( \frac{t + 1}{2t + 1} \right) = 1 + t \]