` tan ^(-1). (sqrt( 1+ x^(2) )-1)/x ` का अवकल गुणांक `tan^(-1)` के सापेक्ष ज्ञात कीजिए ।

Explanation

माना ` y_(1) = tan ^(-1) . (sqrt( 1+x^(2) -1))/x ` तथा ` y_(2) = tan^(-1) x `
` x = tan theta ` पर रखने पर
` y_(1) = tan ^(-1) .(sqrt(1+x^(2))-1)/x = tan^(-1). (sqrt(1 + tan ^(2) theta - 1))/( tan theta) `
` = tan ^(-1). ( sec theta -1)/( tan theta) = tan^(-1). (1- cos theta)/(sin theta) = tan^(-1) .(2 sin ^(2)*1/2 theta)/( 2 sin* 1/2theta cos 1/2 theta)`
` = tan^(-1) (tan *theta/2) = theta/2 = (tan^(-1)x)/2`
` :. (dy_(1))/(dx) = d/(dx) [ tan^(-1) *sqrt(1+x^(2)-1)/x ]`
` = d/(dx) ( (tan^(-1)x)/2) = 1/(2(1+x^(2)))`
तथा ` (dy_(2))/(dx_(2)) = (d[tan^(-1) {sqrt(1-x^(2))-1}//x])/(d(tan^(-1)x))`
` (1/(2(1+x^(2))))/(1/(1+x^(2)))=1/2*`