Question

How do you differentiate `y = ((2x - 1)^3)((x + 1)^3)`?

Answer 1

`(dy)/(dx)=3(2x-1)^2(x+1)^2(4x+1)`

Explanation:

Product rule states if `y(x)=g(x)h(x)`

then `(dy)/(dx)=(dg)/(dx)xxh(x)+(dh)/(dx)xxg(x)`

Hence as `y(x)=(2x-1)^3(x+1)^3`

`(dy)/(dx)=3xx(2x-1)^2xx2xx(x+1)^3+3xx(x+1)^2xx1xx(2x-1)^3`

Here we have also used the concept of function of a function and used chain rule for it. As we have differentiated `(2x-1)^3` and `(x+1)^3` w.r.t. `(2x-1)` and `(x+1)`, we must multiply by differential of `(2x-1)` and `(x+1)` w.r.t `x` i.e. by `2` and `1` respectively

So `(dy)/(dx)=6(2x-1)^2(x+1)^3+3(x+1)^2(2x-1)^3`

= `3(2x-1)^2(x+1)^2[2(x+1)+2x-1]`

= `3(2x-1)^2(x+1)^2[2x+2+2x-1]`

= `3(2x-1)^2(x+1)^2(4x+1)`