InterviewSolution
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InterviewSolution
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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 )] ` |
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Answer» 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 ` |
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