Question

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

Answer 1

The product rule gives us:
For a product of two factors, (First and Second)

`(FS)' = F'S+FS'`.

Now, if there are three factors (First Second and Third), then we get

`(FST)' = (FS)'T+(FS)T' = (F'S+FS')T+(FS)T'` which may be written:

`(FST)' = F'ST+FS'T+FST'`.

For `y=x(1-x)^2(x+2)^3`, we start:

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

We can rewrite this:

`y'=(1-x)^2(x+2)^3 - 2x(1-x)(x+2)^3 + 3x(1-x)^2 (x+2)^2`.

Removing common factors and simplifying gives us:

`y'=(1-x)(x+2)^2 (-6x^2-2x+2) = -2(1-x)(x+2)^2 (3x^2+x-1)`.

Final Note:

Because `(1-x) = -(x-1)`, we could rewrite the middle factor as `(x-1)^2`.

Our answer would look like:

`y' = 2(x-1)(x+2)^2 (3x^2+x-1)`,

which is, of course, equivalent to the expression above.