JEE Mains · Maths · STD 12 - 5. continuity and differentiation
The derivative of \(\tan ^{-1}\left(\frac{\sqrt{1+x^{2}}-1}{x}\right)\) with respect to \(\tan ^{-1}\left(\frac{2 x \sqrt{1-x^{2}}}{1-2 x^{2}}\right)\) at \(x=\frac{1}{2}\) is
- A \(\frac{\sqrt{3}}{12}\)
- B \(\frac{\sqrt{3}}{10}\)
- C \(\frac{2 \sqrt{3}}{5}\)
- D \(\frac{2 \sqrt{3}}{3}\)
Answer & Solution
Correct Answer
(B) \(\frac{\sqrt{3}}{10}\)
Step-by-step Solution
Detailed explanation
Let \(f =\tan ^{-1}\left(\frac{\sqrt{1+ x ^{2}}-1}{ x }\right)\) Put \(x =\tan \theta \Rightarrow \theta=\tan ^{-1} x\) \(f =\tan ^{-1}\left(\frac{\sec \theta-1}{\tan \theta}\right)\) \(f =\tan ^{-1}\left(\frac{1-\cos \theta}{\sin \theta}\right)=\frac{\theta}{2}\)…
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