Question

How do you differentiate `g(y) =(y +2 y^3)(2y^2 -3y^3)` using the product rule?

Answer 1

`y'=y(-36y^4+20y^3-12y^2+6y)`

Explanation:

`(y+2y^3)=u`
`(2y^2-3y^3)=v`
Using product rule
`(dy)/dx=V(du)/dx+U(dv)/dx`
`(du)/dx=(1+6y^2);(dv)/dx=(4y-9y^2)`
`Hence`
`(dy)/dx=(2y^2-3y^3)(1+6y^2)+(y+2y^3)(4y-9y^2)`
`=>2y^2+12y^4-3y^3-18y^5+4y^2-9y^3+8y^4-18y^5`
Collecting like terms
`=>-36y^5+20y^4-12y^3+6y^2`
`=>y(-36y^4+20y^3-12y^2+6y)`
Hope it helps