InterviewSolution
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InterviewSolution
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फलन `sqrtx` का x के सापेक्ष प्रथम सिद्धांत से अवकलन गुणांक ज्ञात कीजिएः। |
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Answer» माना `y=sqrtx=(x)^(1//2)` `therefore y+deltay=(x+deltax)^(1//2)` अंत: `y+deltay-y=(x+deltax)^(1//2)-(x)^(1//2)` `Rightarrow deltay=x^(1//2) [1+(deltax)/(x)]^(-1//2)-x^(1//2)=x^(1//2)[(1+(deltax)/(x))^(1//2)-1]` `Rightarrow deltay=x^(1//2) [1+(1)/2((deltax)/(x))]+((1)/(2)((1)/(2)-1))/(2!)((deltax)/(x))^(1//2)+....oo-1]` `[therefore` द्विपद प्रमेय से, `(1+x)^(n)=1+nx+(n(n-1))/(2!)x^(2)+....]` `Rightarrow deltay=x^(1//2)((delta)/(x))[1+(((1)/(2)-1))/(2!)((deltax)/(x))+....oo]` `=(1)/(2sqrtx)deltax[1+(((1)/(2)-1))/(2!)((deltax)/(x))+......]` दोनों पक्षों को `deltax` से भाग लेकर `underset(x to 0)lim` लेने पर `Rightarrow underset(deltax to o)lim (deltay)/(deltax)=(dy)/(dx)=underset(deltax to 0)lim (1)/(2sqrt2) (deltax)/(deltax) [1+((1)/(2)-1))/(2!) ((deltax)/(x))+....]` `=(1)/(2sqrtx)[1+0+0+.....]` `therefore (d)/(dx) (x^(1//2))=(1)/(2sqrtx)` |
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