COMEDK · Maths · 26. Differentiation
\(\text { If } f(x)=\sin ^{-1}\left(\dfrac{2^{x+1}}{1+4^x}\right) \text { then } f^{\prime}(0) \text { is equal to }\)
- A \(\dfrac{2}{3} \log 2\)
- B \(\log 2\)
- C 0
- D \(2 \log 2\)
Answer & Solution
Correct Answer
(B) \(\log 2\)
Step-by-step Solution
Detailed explanation
Given \(f(x) = \sin^{-1}\left(\dfrac{2^{x+1}}{1+4^x}\right)\). Let \(2^x = \tan \theta\). Then \(4^x = (2^x)^2 = \tan^2 \theta\). The expression inside the inverse sine becomes \(\dfrac{2 \cdot 2^x}{1 + (2^x)^2} = \dfrac{2 \tan \theta}{1 + \tan^2 \theta} = \sin(2\theta)\). Thus,…
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