निम्न फलनों का x का सापेक्ष अवकल गुणांक ज्ञात कीजिए|
` (i) tan ^(-1) [(sin x) /( 1+ cos x }]`
` (ii) tan ^(-1) [ sqrt ((1+ cos x )/( 1- cos x ))]`
`(iii) tan ^(-1)[(sin x +cos x )/( cos x- sin x )] `

Explanation

माना ` y= tan ^(-1) [(sin x)/( 1cos x ) ]` ltbr gt ` rArr " "y= tan ^(-1) [(2sin ( x)/(2) cos ""( x)/(2) ) /(1+2cos ^(2)""( x)/(2) -1)]`

` =tan ^(-1) [tan ((x)/( 2 ) )] =(x)/(2)`
` " " ( because tan ^(-1) (tan theta ) =theta ) `
` therefore " "(dy)/(dx) =(1)/(2)`
(ii) माना ` y=tan ^(-1) [ sqrt ((1+cos x ) /( 1- cos x ))]`
` " " y=tan ^(-1) [(1+2cos ^(2) (x//2)-1 ) /(1-1+2sin ^(2)(x//2))]`
` =tan ^(-1) cot (x//2) `
` =tan ^(-1) tan ""((pi )/(2)- ( x)/(2) )= (pi)/(2) -(x)/(2) `
` " "(because tan ^(-1) (tan theta )= theta ) `
` therefore (dy)/(dx) =-(1)/(2) `
(iii) माना `y= tan ^(-1)[ ( sin x + cos x ) /(cos x - sin x )]`
दाँयें पक्ष में अंश व् हर को cos x से भाग देने पर
` y= tan ^(-1) [(1+tan x) /( 1- tan x) ] `
` = tan ^(-1) tan ""((pi)/(4) +x ) =(pi)/( 4) +x `
` therefore" "(dy)/(dx) =(d)/(dx) ((pi) /(4) +x) =1 rArr (dy)/(dx) = 1 `