Differentiate the following with respect to x.
(i) `(x^3 +2x^2 - 4)(4x^5 - x^2 +1)` (ii) `(7x^5)/(x+3)`

Explanation

(i) `y = (x^3 +2x^2 -4) (4x^5 - x^2 +1)`
`(dy)/(dx) =(x^3 +2x^2 -4) (d)/(dx) (4x^5 -x^2 +1) +(4x^5 -x^2 +1) (d)/(dx) (x^3 + 2x^2 -4)`
`=(x^3+2x^2 - 4) [ 20x^4 - 2x] + (4x^5 - x^2 +1) [ 3x^2 +4x]`
`(20 x^7 +40x^6 - 80x^4 -2x^4 - 4x^3 +8x) +(12x^7 - 3x^4+3 x^2 +16x^6 -4x^3 +4x)`
` = 32 x^7 +56x^6 -85x^4 8x^3 +3x^2 + 12x`
(ii) `y = (7x^5)/((x+3))`
`(dy)/(dx) = ((x+3) (d)/(dx) (7x^5) - 7x^5 (d)/(dx)(x+3))/((x+3)^2) = ((x+3) xx35x^4- 7x^5 (1+0))/((x+3)^(2))`
`= (35x^(5)+105x^(4)-7x^(5))/((x+3)^(2))= (28x^(5) +105x^4)/((x+3)^(2)) = (7x^4(4x +15))/((x+3)^(2)`