निम्न फलन का x के सापेक्ष अवकलन कीजिए|
` y= cot ^(-1) [sqrt((1+x^(2) )+1)/(x)]`

Explanation

` y= cot ^(-1) [ sqrt((1+x^(2)) + 1)/(x) ]`
तब `x= tan theta rArr theta =tan ^(-1) x `
` rArr " "y= cot ^(-1) [sqrt ((1+tan ^(2)theta )+1)/(tan theta )]`
` =cot ^(-1) [(sec theta +1)/(tan theta )]=cot^(-1) ""((1+cos theta )/(sin theta ) ) `
` y= cot ^(-1) [ (1+2cos ^(2)""(theta )/(2)-1)/(2 sin ""(theta )/(2) cos ""(theta )/(2))]=cot ^(-1) (cot ""(theta )/(2))`
` rArr y= (theta)/(2) =(1)/(2) tan ^(-1) x `
` rArr (dy)/(dx) =(1)/(2) (d)/(dx) tan ^(-1) x = (1)/(2(1+x^(2)))`
` therefore " "(dy)/(dx) =(1)/(2(1+x^(2) ))`