InterviewSolution
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InterviewSolution
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निम्न फलन का x के सापेक्ष अवकलन कीजिए| ` (i) y= tan ^(-1) ((sqrt(1+x^(2) -1))/x)` ` (ii) y=sin [2tan ^(-1) sqrt(((1-x)) /((1+x))` |
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Answer» (i ) ` y= tan ^(-1) [(sqrt(1+x^(2) )-1)/(x) ]` माना ` x= tan theta rArr theta tan ^(-1) x ` ` therefore " "y= tan ^(-1) [(sqrt((1+ tan ^(2) theta ))-1)/(tan theta )] =tan ^(-1) [(sectheta -1)/(tan theta )]` ` rArr " "y= tan ^(-1) [(1-cos theta )/(sin theta ) ] =tan ^(-1) tan((theta )/(2))=(theta )/(2)` ` rArr " "y= (1)/(2) tan ^(-1) x ` ` rArr " "(dy)/(dx) =(1)/(2) (d)/(dx) tan ^(-1) x=(1)/(2) ((1)/(a+x^(2)))` ` therefore (dy)/(dx) =(1)/(2(1+x^(2)))` (ii) ` " "y= sin [2tan ^(-1) sqrt(((1-x))/((1+x)))]` माना ` x= cos theta rArr theta cos ^(-1) x ` ` therefore " "=sin [2tan ^(-1)sqrt(((1-cos theta )/(1+costheta )))] ` ` =sin [ 2tan ^(-1) sqrt(((1-1+2sin ^(2)""(theta)/(2))/(1+2cos ^(2) ""(theta )/(2) -1)))]` ` =sin [2tan ^(-1) (tan ""(theta )/(2))] =sin theta ` ` rArr y=sin theta =sin (cos ^(-1) x )` ` therefore (dy)/(dx) =(d)/(dx) sin (cos ^(-1) x ) =cos (cos ^(-1) x ) ((-1)/(sqrt(1-x^(2))))` ` therefore (dy)/(dx) =(-x) /(sqrt((1-x^(2) ) ))` |
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