Question

If $$ y = \sin^{-1} \left( \frac{2x}{1+x^2} \right) + \sec^{-1} \left( \frac{1+x^2}{1-x^2} \right) $$ then the value of $$ \frac{dy}{dx} $$ at $$ x = \sqrt{3} $$ is

• A. \(1\)
• B. \(1/2\)
• C. \(0\)
• D. \(1/4\)
• E. None of these

Hint:

In differentiation questions, always check first whether the function is:
• explicit,
• implicit,
• or parametric. e} Only after knowing the full equation can \(\frac{dy}{dx}\) be evaluated at a specific value of \(x\).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Substitution for inverse trig.
Step 2: Key Formula or Approach:
For \(x \gt 1\), \(\sin^{-1}(2x/(1+x^2)) = \pi - 2\tan^{-1}x\).
Step 3: Detailed Explanation:
\(\sec^{-1}((1+x^2)/(1-x^2)) = \cos^{-1}((1-x^2)/(1+x^2))\).
For \(x \gt 1\), this is \(2\tan^{-1}x\).
Total \(y = \pi - 2\tan^{-1}x + 2\tan^{-1}x = \pi\). Derivative is \(0\).
Step 4: Final Answer:
Result is \(0\).