Obtain the differential coefficient of the following :
`(x-1)(2x+5)`

Answer

(i) `4x + 3`
(ii) `- (2)/((2x+1)^(2))`
(iii) `-(1)/((4x+5)^(2))`
(iv) `(2x-x^(4))/((x^(3)+ 1)^(2))`

Explanation

(i) `(d)/(dx) (x-1)(2x+5)`
`" "=(x-1)(d)/(dx) (2x+ 5) + (2x+ 5) (d)/(dx) (x-1)`
`= (x-1)(2) + (2x+5) (1) = 2x -2+2x+5`
`" "=4x + 3`
(ii) `(d)/(dx) (1)/((2x+1))= (0-2)/((2x+1)^(2))= (-2)/((2x+1)^(2))`
(iii) `(d)/(dx) ((3x+4)/(4x+5)) = ((4x+5)(3)- (3x+4)(4))/((4x+5)^(2))`
`= (-1)/((4x+5)^(2))`
(iv) `(d)/(dx) ((x^(2))/(x^(3)+1)) = ((x^(3)+1)(2x)-x^(2)(3x^(2)))/((x^(3)+1)^(2))`
`= (2x^(4) + 2x -3x^(4))/((x^(3)+ 1)^(2)) = (2x- x^(4))/((x^(3)+ 1)^(2))`