निम्नलिखित के अवकलज ज्ञात कीजिए : (i) `2x-(3)/(4)`
(ii) `(5x^3+3x-1)(x-1)`
(iii) `x^-3(5+3x)`
(iv) `x^5(3-6x^-9)`
(v) `x^-4(3-4x^-5)`
(vi) `(2)/(x+1)-(x^2)/(3x-1)`

Explanation

(i) माना `y=2x-(3)/(4)`
`rArr (dy)/(dx)(x)(2x-(3)/(4))`
`=2(d)/(dx)(x)-(d)/(dx)((3)/(4))`
`=(1)-0=2`
(ii) माना `y=(5x^3+3x-1)(x-1)`
`rArr (dy)/(dx)=(d)/(dx)[(5x^3+3x-1)(x-1)]`
`=(5x^3+3x-1)(d)/(dx)(x-1)+(x-1)(d)/(dx)(5x^3+3x-1)`
`=(5x^3+3x-1)(1-0)+(x-1)(15x^2+3)`
`=5x^3+3x-1+15x^2-3`
`=20x^3-15x^2+6x-4`
(iii) माना `y=x^(-3)(5+3x)`
`=5x^(-3)+3x^(-2)`
`rArr (dy)/(dx)=(d)/(dx)x^(-3)+3x^(-2))`
`=5(d)/(dx)x^(-3)+3(d)/(dx)x^(-2)`
`=5(-3)x^(-4)+3(-2)x^(-3)`
`=-(15)/(x^4)-(6)/(x^(3))=-(3)/(x^3)((5)/(x)+2)`
(iv) माना `y=x^(5)(3-6x^-9)`
`rArr(dy)/(dx)=(d)/(dx)(3x^(5)+6x^(-4))`
`=3(d)/(dx)x^5-6(d)/(dx)x^-4`
`=3(5x^4)-6(-4x^-5)`
`=(15x^4+(24)/(x^5))`
(v) माना `y=x^-4(3-4x^-5)=3x^-4-4x^-9`
`rArr (dy)/(dx)=(d)/(dx)(3x^(-4)-4(d)/(dx)x^(-9))`
`=3(-4x^-5)-4(-9x^-10)`
`=-(12)/(x^5)+(36)/(x^10)=-(12)/(x^5)(1-(3)/(x^5))`
(vi) माना `y=(2)/(x+1)-(x^2)/(3x-1)`
`rArr(dy)/(dx)=(d)/(dx)[(2)/(x+1)-(x^2)/(3x-1)]`
`=(d)/(dx)((2)/(x+1))-(d)/(dx)((x^2)/(3x-1))`
`=((x+1)(d)/(dx)2-2(d)/(dx)(x+1))/((x+1)^2)`
`-((3x-1)(d)/(dx)x^2-x^2(d)/(dx)(3x-1))/((3x-1)^2)`
`=(0-2)/((x+1)^2)-((3x-1).2x-x^2 .3)/((3x-1)^2)`
`=(-2)/((x+1)^2)-(3x^2-2x)/((3x-1)^2)`
`=-(2)/((x+1)^2)-(x(3x-2))/((3x-1)^2)`