निम्न फलनों का x के सापेक्ष अवकल गुणांक ज्ञात कीजिये
` (i) (tan ^(-1) x ) /( x ) `
` (ii) x sec ^(-1) x ^(3) `
` (iii) (sin ^(-1) x )^(m)`

Explanation

(i ) माना ` y= (tan ^(-1) x ) /( x ) `
` therefore (dy)/(dx) =( x(d)/(dx) tan ^(-1) x-tan ^(-1) x (d)/(dx) x)/( x^(2) ) `
` " "( ( x)/( 1+x^(2) )- tan ^(-1) x)/( x^(2) )`
या ` (dy)/(dx) =(x -(1+x^(2) )tan ^(-1) x)/( x^(2) (1+x^(2) ) ) `
(ii) माना ` y = sec ^(-1) x^(3) `
` therefore " "(dy)/(dx) =x (d)/(dx) sec ^(-1) x^(3) + sec^(-1)x^(3) (d)/(dx) x `
` = x(1xx3x^(2) )/( x^(3) sqrt (x^(6)) -1)+ sec ^(-1) x^(3) `
` = (3)/(sqrt (x^(6)) -1 ) +sec ^(-1) x^(3)`
माना ` y= (sin ^(-1) x)^(m) `
` therefore " "(dy)/(dx) =(d)/(dx) (sin^(-1) x )^(m) `
` " "=m(sin ^(-1) x ) ^(m-1) (d)/(dx) sin ^(-1) x `
` rArr" "(dy)/(dx) =(m (sin ^(-1) x) ^(m-1))/(sqrt(1-x^(2)))`