InterviewSolution
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InterviewSolution
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`(sin x)/(x)` का प्रथम सिद्धांत से अवकल गुणांक ज्ञात कीजिएः |
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Answer» माना `y=(sin x)/(x)` `Rightarrow y+deltay=(sin(x+deltax))/(x+deltax)` `Rightarrow y+deltay-y=(sin (x+deltax))/(x+delta)-(sin x)/(x)` `Rightarrow deltay=(sin(x+deltay))/(x+deltay)-(sin x)/(x)` `=(x sin(x+deltax)-(x+deltax)sinx)/(x(+deltax))` `Rightarrow (deltay)/(deltax)(x sin(x+deltax)-(x+deltax)sinx)/(deltax.x(x+deltax))` `therefore (dy)/(dx)=underset(deltax to 0)lim (deltay)/(deltax)` `=underset(deltax to 0)lim ({x sin (x+deltax)-(x+deltax)sin x})/(x(x+deltax).deltax)` `=underset(deltax to 0)lim (x[ sin (x+deltax)-sin x]-deltax.sin x)/(x(x+deltax).deltax)` `=underset(deltax to 0)lim (x[(2cos (x+(deltax)/(2))sin((deltax)/(2)))]-(deltax)sin x)/(x(x+deltax).deltax)` `underset(deltax to 0)lim cos (x+(deltax)/(2)),underset(deltax to 0)lim (sin(deltax//2))/((deltax//2)).underset(deltax to 0)lim (1)/((x+deltax))-underset(deltax to 0)lim (sin x)/(x(x+deltax))` `=(cos.x.(1)/(x))-(sin x)/(x^2)` `=(cos x)/(x)-(sin x)/(x^(2))` `=(x cos x-sin x)/(x^(2))` |
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