फलन `x^(-6)` का `x` के सापेक्ष प्रथम सिद्धांत से अवकलन गुणांक ज्ञात कीजिएः

Explanation

माना `y=x^(-6)` ......(i)
`Rightarrow y+deltay=(x+deltax)^(-6)` .........(ii)
`therefore y+deltay-y=(x+deltax)^(-6)-x^(-6)`
`Rightarrow deltay=x^(-6) (1+(deltax)/(x))^(-6)-x^(-6)`
`=x^(-6) [(1+(deltax)/(x))^(-6)-1]`
`Rightarrow deltay=x^(-6)[1+(-6)((deltax)/(x))+((-6)(-6-1))/(2!) ((deltax)/(x))^(2)+......oo-1]`
`Rightarrow deltay=x^(-6)[(-6)((deltax)/(x))+((-6)(-7))/(2!) ((deltax)/(x))^(2)+......oo]`
`=x^(-6)(-6)((deltax)/(x))[1+((-7))/(2!) ((deltax)/(x))+......oo]`
दोनों पक्षों को `deltax` से भाग देकर `underset(deltax to 0)` लेने पर,
`underset(deltaxx to 0)lim (deltay)/(deltax)=(dy)/(dx)=underset(deltax to 0)lim -6x^(-7)(deltax)/(deltax)[1+((-7))/(2!) ((deltax)/(x))+......oo]`
`=underset(deltax to 0)lim -6x^(-7)[1+((-7))/(2!) ((deltax)/(x))+......oo]`
`=-6x^(-7)[1+0+0+......oo]`
`therefore (d)/(dx)(x^(-6))=-6x^(-7)=-(6)/(x^(7))`