Limits, Continuity and Differentiability
80304
If \(f(x)=\sin ^{-1}\left(\sqrt{\frac{1-x}{2}}\right)\), then \(f^{\prime}(x)=\)
1 \(\frac{1}{2 \sqrt{1+\mathrm{x}^{2}}}\)
2 \(\frac{-1}{2 \sqrt{1-x^{2}}}\)
3 \(\frac{1}{\sqrt{1-\mathrm{x}^{2}}}\)
4 \(\frac{-1}{2 \sqrt{1+\mathrm{x}^{2}}}\)
Explanation:
(B) : Given,
\(f(x)=\sin ^{-1}\left(\sqrt{\frac{1-x}{2}}\right)\)
Let, \(\quad \mathrm{x}=\cos 2 \theta \quad \theta=\frac{1}{2} \cos ^{-1} \mathrm{x}\)
Then,
\(f(x)=\sin ^{-1}\left(\sqrt{\frac{1-\cos 2 \theta}{2}}\right)\)
\(f(x)=\sin ^{-1}\left(\sqrt{\frac{1-1+2 \sin ^{2} \theta}{2}}\right)\)
\(f(x)=\sin ^{-1}(\sin \theta) \Rightarrow f(x)=\theta\)
On putting the value \(\theta\) in above equation,
\(f(x)=\frac{1}{2} \cos ^{-1} x\)
On differentiating both sides. w.r.t.x
\(\mathrm{f}^{\prime}(\mathrm{x})=\frac{-1}{2 \sqrt{1-\mathrm{x}^{2}}}\)