Differentiate the following w.r.tx
(i) `(5x^(2) +6)(2 x^(3) +4 )` (ii) `sqrtx (x^(2) +7)` (iii) `(x^(2) +3 )(x^(4) - 9)`

Answer

(i) `50x^4 +36 x^2 +40 x` (ii) `(5)/(2)x^(3//2) +(7)/(2) x^(-1//2)` (iii) `6x (x^4 + 2x^2 -3)`

Explanation

(i) `(dy)/(dx) = (5x^2+6)(d)/(dx) (2x^3 +4)+(2x^3 +4)(d)/(dx)(5x^2 +6)xx6x^2 +(2x^3 +4)xx10x`
`=30 x^4 +36x^2 +20x^4 +40x = 50x^4 +36x^2 +40x`
(ii)`(dy)/(dx) =x^(1)/(2) (d)/(dx)(x^2 +7)+(x^2+7)(d)/(dx)(x^(1)/(2))(2x +0)+(x^2 +7)((1)/(2)x^(-1/2))`
`= 2x^(3//2) +(1)/(2)x^(3//2) +(7)/(2) x^(1/2) = (5)/(2) x^(3//2) + (7)/(2)x^(-1//2)`
(iii) `(dy)/(dx) = (x^2 +3)(d)/(dx) (x^4 - 9)+(x^4 - 9)(d)/(dx)(x^2+3) = (x^2 +3)[4x^3 - 0]+(x^4 - 9)[2x-0]`
`=4x^5 +12x^3 +2x^5 - 18x = 6x^5 +12x^3 - 18 x =6x[x^4 +2x^2 -3]`