What are the roots of the derivative of y=x^2/(2x-1)-2/(2x-1) ?

We'll have to differentiate the function for finding the
roots of the expression of the first derivative.

We'll
re-write the function:

f(x) = (x^2 -
2)/(2x-1)

Since the given function is a product, we'll
apply the quotient rule:

f'(x) = [(x^2-2)'*(2x-1)
- (x^2-2)*(2x-1)']/(2x-1)^2

f'(x) = [2x(2x-1) - 2(x^2 -
2)]/(2x-1)^2

We'll remove the
brackets:

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

We'll combine like
terms:

f'(x) = (2x^2 - 2x +
4)/(2x-1)^2

Now, we'll put f'(x) =
0.

(2x^2 - 2x + 4)/(2x-1)^2 =
0

Since the denominator is always positive, for any value
of x, only the numerator could be zero.

2x^2 - 2x + 4 =
0

We'll calculate delta:

delta
= b^2 - 4ac

We'll identify
a,b,c:

a = 2 , b = -2 , c =
4

delta = 4 - 32 = -28 <
0

Since delta is negative and a = 2>0, the
expression 2x^2 - 2x + 4 is always positive for any avlue of
a.

So, the first derivative is positive and
it is not cancelling for any value of a.

The equation f'(x) = 0 has no
roots.