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Find the derivative of (i) 2x-3/4 (ii) ...

Find the derivative of (i) `2x-3/4` (ii) `(5x^3+3x-1)(x-1)` (iii) `x^(-3)(5+3x)`

Text Solution

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(i) Let `y=2x-(3)/(4)`
`rArr (dy)/(dx)=(d)/(dx)(2x-(3)/(4))`
`=2(d)/(dx)(x)-(d)/(dx)((3)/( 4))`
`=2(1)-0=2`
(ii) Let `y=(5x^(3)+3x-1)(x-1)`
`rARr (dy)/(dx)=(d)/(dx)[(5x^(3)+3x-1)(x-a)]`
`=(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^(3)+3x-15x^(2)-3`
`=20x^(3)-15x^(2)+6x-4`
(iii) Let `y=x^(-3)(5+3x)`
`=5x^(-3)+3x^(-2)`
`rArr (dy)/(dx)=(d)/(dx)(5x^(-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) Let `y=x^(5)(3-6x^(-9))`
`=3x^(5)-6x^(-4)`
`rArr(dy)/(dx)=(d)/(dx)(3x^(5)-6x^(-4))`
`=d(d)/(dx)x^(5)-6(d)/(dx)x^(-4)`
`=3(5x^(4))-6(-4x^(-5))`
`=(15x^(4)+(24)/(x^(5)))`
(v) Let `y=x^(-4)(3-4x^(-5))=3x^(-4)-4x^(-9)`
`rArr(dy)/(dx)=(d)/(dx)(3x^(-4)-4x^(-9))`
`=3(d)/(dx)x^(-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) Let `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))-(1)/(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))`
`=-(1)/((x+1)^(2))=(x(3x-2))/((3x-1)^(2))`.
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