`sqrt((3x-4))` का `x` के सापेक्ष प्रथम सिद्धांत से अवकल गुणांक ज्ञात कीजिएः

Options

A. `(3)/(sqrt((3x-4)))`
B. `(3)/(7sqrt((3x-4)))`
C. `(3)/(5sqrt((3x-4)))`
D. `(3)/(2sqrt((3x-4)))`

Answer

D

Explanation

`y=sqrt(3x-4)=(3x-4)^(1//2)` .......(i)
`Rightarrow y+deltay=[3(x+deltay)-4]^(1//2)`
`y+deltay=[(3x-4)+3deltax]^(1//2)........(ii)`
`y+deltay-y=[(3x-4)+3deltax]^(1//2)-(3x-4)^(1//2)`
` =(3x-4)^(1//2) [(1+(3deltax)/(3x-4))^(1//2)]-(3x-4)^(1//2)`
` Rightarrow deltay=(3x-4)^(1//2) [(1+(3deltax)/(3x-4))^(1//2)-1]`
` Rightarrow deltay=(3x-4)^(1//2) [1+(1)/(2)((3deltax)/(3x-4))+((1)/(2)((1)/(2)-1))/(2!)((3deltax)/(3x-4))^2+.....oo-1]`
` =(3x-4)^(1//2) [(1)/(2)((3deltax)/(3x-4))+((1)/(2)((1)/(2)-1))/(2!)((3deltax)/(3x-4))^2+.....oo]`
` =(3x-4)^(1//2) .(1)/(2)(3deltax)/(3x-4)[1+((1)/(2)((1)/(2)-1))/(2!)((3deltax)/(3x-4))+.....oo]`
` =(3deltax)/(2(3x-4)^(1//4))[1+((1)/(2)((1)/(2)-1))/(2!)((3deltax)/(3x-4))+.....oo]`
दोनों पक्षों `deltax` को से भाग देकर `underset(deltax to 0)` लेने पर,
`underset(deltax to 0)lim (deltay)/(deltax)=(dy)/(dx)=underset(deltax to 0)lim (3)/(2(3x-4)^(1//2))=(deltax)/(deltax)[1+(((1)/(2)-1))/(2!)(3deltax)/((3x-4))+......oo]`
`=3/(2(3x-4)^(1//2)) [1+0+0+.....]`
`=(3)/(2sqrt((3x-4)))`
`therefore (d)/(dx)(3x-4)^(1//2)=(3)/(2sqrt((3x-4)))`