- #1
Lo.Lee.Ta.
- 217
- 0
Hi, I got the right answer when I used the Quotient Rule but not when I used the Product Rule...
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!
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!