Solveeit Logo
Login

Question

Mathematics Question on Functions

If f(x) = sin⁡−1\sin^{-1} (2x1+x2)\left(\frac{2x}{1+x^{2}}\right), then f' (3)(\sqrt{3}) is

A

−12-\frac {1}{2}

B

12\frac {1}{2}

C

13\frac {1}{\sqrt{3}}

D

−13-\frac {1}{\sqrt{3}}

Answer

12\frac {1}{2}

Explanation

Solution

f(x)=sin⁡−1(2x1+x2)f(x)=\sin ^{-1}\left(\frac{2 x}{1+x^{2}}\right)
Put x=tan⁡ θx=\tan\, \theta, where θ∈(−π2,π2)\theta \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)
f(x)=sin⁡−1(2tan⁡ θ1+tan⁡2θ)f(x)=\sin ^{-1}\left(\frac{2 \tan \,\theta}{1+\tan ^{2} \theta}\right)
⇒f(x)=sin⁡−1(sin⁡2θ)\Rightarrow f(x)=\sin ^{-1}(\sin 2 \theta)
⇒f(x)=2 θ=2tan⁡−1x  (∵θ=tan⁡−1x)\Rightarrow f(x)=2 \,\theta=2 \tan ^{-1} x \,\,\left(\because \theta=\tan ^{-1} x\right)
f′(x)=21+x2f'(x)=\frac{2}{1+x^{2}}
∴f′(3)=21+3=12\therefore f'(\sqrt{3})=\frac{2}{1+3}=\frac{1}{2}