Question

Find the derivative and simplify:
f(x)= [x(x^2-1)]/[x+3]
I'm not sure if I should use the Product Rule, the Chain Rule, the Quotient Rule, or a combination of two!!
I would love some help and 'thank you' in advance!

Answer 1

looks like you use all of the above

Answer 2

I looked at the work I did for this and I honestly don't think I did it correctly. I believe I used the product rule and then the quotient rule?
I got something like this:
f'(x)= (2x^3+9x^2-3)/(x^2+9x+9)

Answer 3

I would not expand the factors...

you could start with the quotient rule
\[ d \frac{u}{v}= \frac{v\ du - u\ dv }{v^2} \]
where u is x(x^2-1) and v is (x+3)

to find du you use the product rule
d( a b) = a db + b da
where x will be a and (x^2+1) is b
to find db you find d(x^2 + 1) and use the power rule.

Answer 4

so you think I should do the quotient rule and then the product rule? I think I did it in the opposite order.
I'll try working this out, then.

Answer 5

But if my earlier answer was correct, you meant to not expand and keep it like this:
(2x^3 +9x^2 -3)/(x+3)^2
?

Answer 6

yes, I would not expand that out unless necessary (for example, multiple choice had the answers in expanded form)

Answer 7

also, for this problem, it might be easier to expand x(x^2 -1) to x^3 - x
so that you don't have to use the product rule.
(or do it both ways, as a check. You should get the same answer)

Answer 8

ok, I'll try that now to check my work!

Answer 9

ok, so I was correct! yay! I was just a little unsure of myself.
Thank you for your help!
I'll award Best Response if I can, but I'm not sure if I have any medals...

Answer 10

btw, your answer up above
****
I got something like this:
f'(x)= (2x^3+9x^2-3)/(x^2+9x+9)
*****
looks ok, except the bottom should be (x+3)^2 = x^2 +6x+9

It looks like you should expand the terms up top, because it simplifies down quite a bit.

Answer 11

oh ok so if I have (2x^3 +9x^2 -3)/(x^2+6x+9)
I didn't see any way to simplify it...? no common factors as far as I can tell

Answer 12

that looks good.

Answer 13

ok then! thank you for your time and help! i'll close this question.

Answer 14

I was thinking that after we take the derivative, we could leave the top "as is", but it makes sense to do what you did, and rewrite it as 2x^3 +9x^2 -3

Answer 15

ok that's what I had thought. my calc teacher likes things simplified, but not too simplified. :3

Answer 16

though the bottom I would leave as (x+3)^2
it looks nicer that way

Answer 17

alright, well I have what you just said as the answer and I have it expanded underneath, just in case!