Question

what is the instantaneous rate of change at x = 2 of the function f given by f(x)=\frac{x^{2}-1}{x - 1}? a) 2 b) \frac{1}{2} c) \frac{1}{4} d) 2 e) 6

Explanation:

Step1: Recall derivative formula

The derivative of $y = \frac{u}{v}$ is $y'=\frac{u'v - uv'}{v^{2}}$ (quotient - rule). For $f(x)=\frac{3x - 1}{x - 1}$, let $u = 3x - 1$, then $u'=3$; let $v=x - 1$, then $v' = 1$.

Step2: Apply quotient - rule

$f'(x)=\frac{3(x - 1)-(3x - 1)\times1}{(x - 1)^{2}}=\frac{3x-3 - 3x + 1}{(x - 1)^{2}}=\frac{-2}{(x - 1)^{2}}$.

Step3: Evaluate derivative at $x = 2$

Substitute $x = 2$ into $f'(x)$. $f'(2)=\frac{-2}{(2 - 1)^{2}}=-2$.

Answer:

There is no correct option provided among A, B, C, D, E as the answer is - 2.