To find the derivative of the function [tex]\( y = \frac{x-1}{2x^2 - 7x + 5} \)[/tex] with respect to [tex]\( x \)[/tex] and then evaluate this derivative at [tex]\( x = 2 \)[/tex], we can follow these steps:
1. Identify the function and its components:
Given function:
[tex]\[
y = \frac{x-1}{2x^2 - 7x + 5}
\][/tex]
2. Apply the quotient rule:
The quotient rule for differentiation states:
[tex]\[
\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{u'v - uv'}{v^2}
\][/tex]
where [tex]\( u = x - 1 \)[/tex] and [tex]\( v = 2x^2 - 7x + 5 \)[/tex].
3. Compute [tex]\( u' \)[/tex] and [tex]\( v' \)[/tex]:
[tex]\[
u = x - 1 \implies u' = 1
\][/tex]
[tex]\[
v = 2x^2 - 7x + 5 \implies v' = 4x - 7
\][/tex]
4. Apply the quotient rule with these components:
[tex]\[
\frac{dy}{dx} = \frac{(1)(2x^2 - 7x + 5) - (x-1)(4x - 7)}{(2x^2 - 7x + 5)^2}
\][/tex]
5. Simplify the expression:
Expand the numerator:
[tex]\[
\text{Numerator} = (2x^2 - 7x + 5) - (x-1)(4x - 7)
\][/tex]
Expand [tex]\( (x-1)(4x - 7) \)[/tex]:
[tex]\[
(x-1)(4x - 7) = 4x^2 - 7x - 4x + 7 = 4x^2 - 11x + 7
\][/tex]
Subtract this from [tex]\( (2x^2 - 7x + 5) \)[/tex]:
[tex]\[
\text{Numerator} = 2x^2 - 7x + 5 - (4x^2 - 11x + 7)
\][/tex]
[tex]\[
= 2x^2 - 7x + 5 - 4x^2 + 11x - 7
\][/tex]
[tex]\[
= -2x^2 + 4x - 2
\][/tex]
6. Fit it back into the derivative expression:
[tex]\[
\frac{dy}{dx} = \frac{-2x^2 + 4x - 2}{(2x^2 - 7x + 5)^2}
\][/tex]
7. Evaluate at [tex]\( x = 2 \)[/tex]:
Substituting [tex]\( x = 2 \)[/tex] into the derivative:
[tex]\[
\frac{dy}{dx} \bigg|_{x=2} = \frac{-2(2)^2 + 4(2) - 2}{(2(2)^2 - 7(2) + 5)^2}
\][/tex]
[tex]\[
= \frac{-2(4) + 8 - 2}{(8 - 14 + 5)^2}
\][/tex]
[tex]\[
= \frac{-8 + 8 - 2}{(-1)^2}
\][/tex]
[tex]\[
= \frac{-2}{1}
\][/tex]
[tex]\[
= -2
\][/tex]
Therefore, the derivative [tex]\( \frac{dy}{dx} \)[/tex] at [tex]\( x = 2 \)[/tex] is [tex]\( -2 \)[/tex].
