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

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Answer 1 · 2015-04-03T11:34:31

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+x2(1-x)(-1)^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.

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