Question

If \( \tan^{-1} x = y \), then \( \frac{dy}{dx} \) is equal to :

• A. \( (\sec^{-1} x)^2 \)
• B. \( \sec^2 y \)
• C. \( \frac{1}{\sqrt{1+x^2}} \)
• D. \( \cos^2 y \)

Hint:

When evaluating derivatives involving inverse trigonometric expressions, if the standard answer \( \frac{1}{1+x^2} \) is not present in the options, look to convert the variable \( x \) back into terms of \( y \) using identities like \( \sec^2 y = 1 + \tan^2 y = 1 + x^2 \). Thus, \( \frac{1}{1+x^2} = \frac{1}{\sec^2 y} = \cos^2 y \).

Answer 1

The Correct Option is D