# Math Help - Compute f'(x) and finding critical numbers

1. ## Compute f'(x) and finding critical numbers

Hey guys

Here's the solution to a question i had on a test

ok so i don't understand how they got from:
$-x^3/(3x-2)(2+3x)$
to
${3x^2(4-3x^2)}/(2-3x)^2(2+3x)^2$

and i don't get how they find the "set numerator to 0" and "set denominator to 0" answers.

any help REALLY appreciated, thanks!

2. This looks like a part of a test or assignment, and it is against MHF Policy to post these..

Read Rule No 6.

3. When you differentiate this:

$f(x) = \dfrac{-x^3}{(3x-2)(2+3x)}$

you get:

$f'(x) = \dfrac{3x^2(4-3x^2)}{(2-3x)^2(2+3x)^2}$

using the quotient rule.

The denominator can be written as:

$(3x-2)(2+3x) = (3x-2)(3x+2) = 9x^2 -4$

and the quotient rule applied:
If
$y = \dfrac{u}{v}$

then;

$y' = \dfrac{v u' - uv'}{v^2}$

When you set the numerator to zero, you are looking for the maxima and minima (and/or inflexion)
When you set the denominator to zero, you are looking for the vertical asymptotes.