Question

If \[ y = \tan^{-1} \left( \frac{1}{x^2 + x + 1} \right) + \tan^{-1} \left( \frac{1}{x^2 + 3x + 3} \right) + \tan^{-1} \left( \frac{1}{x^2 + 5x + 7} \right) + \cdots { (to n terms)} \], then \(\frac{dy}{dx}\) is:

• A. \( \frac{1}{x^2 + n^2} - \frac{1}{x^2 + 1} \)
• B. \( \frac{1}{(x + n)^2 + 1} - \frac{1}{x^2 + 1} \)
• C. \( \frac{1}{x^2 + (n + 1)^2} - \frac{1}{x^2 + 1} \)
• D. None of these

Hint:

For trigonometric series involving inverse functions and variables, look for telescoping patterns to simplify the expression before differentiating.