- #1

Lo.Lee.Ta.

- 217

- 0

I think it might be an algebra mistake...

Product Rule Method:

f'(x) = (3 - x^2)*(4 + x^2)^-1

= (3 - x^2)[(-1(4 + x^2)^-2)*2x] + [(4 + x^2)^-1](-2x)

= [(3 - x^2)(-2x)]/[(4 + x^2)^2] + [(-2x)/(4 + x^2)]

To get the same denominator here, I thought I might square the factor on the right.

= (-6x + 2x^3 + 4x^2)/[(4 + x^2)^2] <----- This is not right.

THE RIGHT ANSWER IS: -14x/[(4 + x^2)^2]

I found that by using the Quotient Rule.

...But where am I going wrong with the Product Rule...?

P.S. Oh, and a side note:

If I wrote out the Product Rule answer like this: (-6x + 2x^3 + 4x^2)/(x^4 + 8x^2 + 16)

Would the 4x^2 and the 8x^2 be able to cancel somewhat?

Sometimes I get confused about what can cancel.

Could it turn into:

(-6x + 2x^3)/(x^4 + 2 + 16) = (-6x + 2x^3)/(x^4 + 18)?

I know that's not the right answer, but is this how it would cancel?

Thank you very much!