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"Find"(dy)/(dx)"if "y=tan^(-1)((sqrt(1+x...

`"Find"(dy)/(dx)"if "y=tan^(-1)((sqrt(1+x^2)-1)/x),` where `x!=0`

Text Solution

Verified by Experts

The correct Answer is:
`(1)/(2)((1)/(1+x^(2)))`
`"Let "y=tan^(-1)((sqrt(1+x^(2))-1)/(x))."Putting "x= tan theta," we get"`
`y=tan^(-1)((sec theta-1)/(tan theta))`
`=tan^(-1)((1-cos theta)/(sin theta))`
`=tan^(-1)((2sin^(2)""(theta)/(2))/(2 sin""(theta)/(2)cos""(theta)/(2)))`
`=tan^(-1)(tan""(theta)/(2))=(1)/(2)theta=(1)/(2)tan^(-1)x`
`therefore" "(dy)/(dx)=(1)/(2)((1)/(1+x^(2)))`
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