InterviewSolution
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InterviewSolution
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यदि ` y= (1+(1)/(x) )^(x) + x^(1+(1)/(x))` तो `(dy)/(dx) ` का मान ज्ञात कीजिए| |
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Answer» दिया है- ` " "y= (1+(1)/(x))^(x)+x ^(1+(1)/(x))` माना ` " "u =(1+(1)/(x) ) ^(x)` तथा ` " "v=x^(1+(1)/(x))` ` rArr " "y=u +v " "rArr(dy)/(dx)= (du)/(dx)+(dv)/(dx)" "...(1)` अब ` " "u= (1+(1)/(x))^(x) rArr log u =x log (1+(1)/(x))` दोनों पक्षों का x के सापेक्ष अवकलन करने पर ` " "(1)/(u) (du)/(dx) =x (d)/(dx) log (1+(1)/(x) )+log (1+(1)/(x))(d)/(dx) (x)` ` " "= x *(1)/(1+(1)/(x))*(d)/(dx) (1+(1)/(x)) +log(1+(1)/(x))*1` `rArr " "(du)/(dx)=u [(x^(2)/x+1)(0-(1)/(x^(2)))+ log (1+(1)/(x))]` ` " "= (1+(1)/(x))^(x) [log (1+(1)/(2))-(1)/(1+x)] " "...(2)` अब ` v=x^(1+(1)/(x))" "rArrlog v =(1+(1)/(x) ) log x ` दोनों पक्षों का x के सापेक्ष अवकलन करने पर ` rArr " "(1)/(v) (dv)/(dx) =(1+(1)/(x))*(1)/(x) +log x*(0-(1)/(x^(2)))` ` rArr " "(dv)/(dx)=v [(1)/(x) +(1)/(x^(2) )-(1)/(x^(2))log x ]` ` " "= x^(1+(1)/(2))[(1)/(x)+(1)/(x^(2))(1-log x )]` समीकरण (1 ),(2 ),व (3 ) से, ` (dy)/(dx) = (1+(1)/(x))^(x) [log (1+(1)/(x))-(1)/(1+x)]` ` " "+ x^(1+(1)/(x))[ (1)/(x)+ (1)/(x^(2))(1-log x )]` |
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