प्रथम सिद्धांत के प्रयोग से `x^(-3//2)` का अवकल गुणांक ज्ञात कीजिएः

Explanation

माना `y=x^(-3//2)`
`Rightarrow y+deltay=(x+deltax)^(-3//2)`
`Rightarrow y+deltay-y=(x+deltax)^(-3//2)-x^(-3//2)`
`Rightarrow (deltay)/(deltax)=((x+deltax)^(-3//2)-x^(-3//2))/(deltax)`
`therefore (dy)/(dx)=underset(deltax to 0)lim (deltay)/(deltax)=underset(deltax to 0)lim [((x+deltax)^(-3//2)-x^(-3//2))/(deltax)]`
`=underset(deltax to 0)lim ([(x+deltax)^(-3//2)-x^(-3//2)])/([(x+deltax)-x))`
`=underset(t to x)lim (t^(-3//2)-x^(-3//2))/(t-x) " "("जहाँ" t=x+deltax)` ltrgt `=-(3)/(2)x^(-5//2)` `(underset(t to a)lim (t^(n)-a^(n))/(t-a)=na^(n-1)` प्रयोग करते हुए)