To find \( f'(2) \) for the given function \( \displaystyle f(x) = \frac{x\, – 2}{x^2 – 3x + 2} \), we will first simplify it and then will find the derivative.
Solution:
\( \displaystyle f(x) = \frac{x\, – 2}{x^2 – 3x + 2} \\ \displaystyle f(x) = \frac{x\, – 2}{(x\, – 1)(x\, – 2)} \\ \)Since \( x \neq 2 \),
\( \displaystyle f(x) = \frac{1}{x\, – 1} \quad \text{for} \quad x \neq 2 \)
Taking derivative with respect to \( x\), we get
\( \displaystyle f'(x) = -\frac{1}{(x – 1)^2} \)
Substitute \( x = 2 \) into the derivative:
\( \displaystyle f'(2) = -\frac{1}{(2 – 1)^2} = -1 \\ \)Therefore, \( \boxed{f'(2) = -1} \)
Please let me know in the comments if you find any error in this solution.