Simple Differentiation Problem

[Vendoskt]

I labeled this topic as a "Simple Differentiation Problem" because I know that it is simple, I'm just having problems with it.

The question is to differentiate ((x^2 + 2)^3)(x - 3)

The answer the book I'm using gives is ((x^2 + 2)^2)(7x^2 - 18x + 2)

Would this be differentiated by using a combination of product rule and chain rule? If that is the case then...

y' = ((x^2 + 2)^3) + (x - 3)(3(x^2 + 2)^2)(2x)

simplified a little

y' = ((x^2 + 2)^3) + 6x(x - 3)((x^2 + 2)^2) or

y' = ((x^2 + 2)^3) + (6x^2 - 18x)((x^2 + 2)^2)

this is where I am getting stuck. Either I am seriously lacking in my algebra skills, or I just used the wrong method to solve the problem.

Please help

Thank you

 

[Hammie]

((x^2 + 2)^3) + (6x^2 - 18x)((x^2 + 2)^2)

you might try factoring a (x^2+2) out of ((x^2+2)^3)

 

[FrogPad]

When in doubt you could expand the entire expression:
$$(x^2+2)^3(x-3)$$

Instead think of it as two terms:
$$A\times B$$
$$A = (x^2+2)^3$$
$$B = (x-3)$$

Thus to differentiate $\frac{d}{dx} [AB] = \frac{d}{dx}(A) B + A\frac{d}{dx}(B)$

So the same logic applies:
$$\frac{d}{dx}\left( (x^2+2)^3(x-3) \right) = \frac{d}{dx} \left[ (x^2+2)^3\right] (x-3) + \frac{d}{dx}\left[(x-3)\right](x^2+2)^3$$

To take care of that $\frac{d}{dx} \left[(x^2+2)^3\right]$ you use the chain rule.

So yup. You use the product and chain rule, or you could expand the entire thing out and do it that way... which might take awhile.

I don't know if you have ever used the program Maple. But it has a really nice tutor that walks you through differentiating. It's pretty cool, and quite helpful for these problems. It doesn't just spit out the correct answer, it instead shows you how to get it.