differentiating a rational quadratic

Hello, I am trying to differentiate this function so I work out where it is increasing or decreasing.

What is the best procedure to do this. I tried using quotient rule but it seemed very laborious. Is there a better way ?

2x^2+x-1 / x^2+x-2

Thanks for any help.

Re: differentiating a rational quadratic

Maybe try factoring each quadratic with the purpose of cancellation. Otherwise your stuck with the quotient rule.

Re: differentiating a rational quadratic

So our function is, .

You need to find .

Try to continue.

Re: differentiating a rational quadratic

thank you. I have tried to continue to simplify this derived function so that I can work out it's roots.

I can see I could split it and cancel one of the (x^2+x-2) factors, but that is as far as I can get.

I really am stuck with this one. Any help proceeding would be great.

Re: differentiating a rational quadratic

as stated by **AsZ**, expand the numerator and combine like terms ... in other words, do the grunt work algebra and finish simplifying the numerator.

You'll end up with a quadratic in the numerator that won't factor, so you'll need to complete the square or use the quadratic formula to find where the derivative equals 0.

Re: differentiating a rational quadratic

my grunt work gives me:

x^2-6x-1 / x^4+2x^3-3x^2-4x+4

I equated the numerator to zero to find the roots. That gave me roots at 6.16 and -0.162 however looking at a graph of 2x^2+x-1 / x^2+x-2 that cannot be correct.

I double checked but to no avail.

If anyone can spot my error that would be appreciated.

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Re: differentiating a rational quadratic

Quote:

Originally Posted by

**fran1942** my grunt work gives me:

x^2-6x-1 / x^4+2x^3-3x^2-4x+4

I equated the numerator to zero to find the roots. That gave me roots at 6.16 and -0.162 however looking at a graph of 2x^2+x-1 / x^2+x-2 that cannot be correct.

I double checked but to no avail.

If anyone can spot my error that would be appreciated.

your roots for the derivative are correct, ... note the relative max at and the relative min at in the graph (marked by the red diamonds)