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Find the derivative of sec^(-1)(1/(2x^2-...

Find the derivative of `sec^(-1)(1/(2x^2-1))wdotrdott` `sqrt(1-x^2)` at `x=1/2`

Text Solution

Verified by Experts

The correct Answer is:
`-4`
Let ` u = sec^(-1)(-1/(2x^(2)-1)) and v=sqrt(1-x^(2))`
Put `" " x = cos theta`
` :." " u=sec^(-1)(-sec 2 theta) and v= sin theta `
` rArr" " u=pi-2 theta" "[:' sec^(-1)(-x)=pi -sec^(-1) x]`
` and " " v = sin theta `
` rArr" "(du)/(d theta) =- 2`
` rArr" " (du)/(d theta) =- 2`
an ` " " d(du)/(dv) =- 2/(cos theta) ,((du)/(dv))_(theta = pi//3) =- 4`
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