Question

find the derivative of $\frac{x + 1}{x - 1}$

Explanation:

Step1: Apply quotient - rule

The quotient - rule states that if $y=\frac{u}{v}$, then $y^\prime=\frac{u^\prime v - uv^\prime}{v^{2}}$. Here, $u = x + 1$, $u^\prime=1$, $v=x - 1$, $v^\prime = 1$.

Step2: Substitute values into formula

$y^\prime=\frac{1\times(x - 1)-(x + 1)\times1}{(x - 1)^{2}}$.

Step3: Simplify the numerator

$y^\prime=\frac{x-1-x - 1}{(x - 1)^{2}}$.

Step4: Further simplify

$y^\prime=\frac{-2}{(x - 1)^{2}}$.

Answer:

$\frac{-2}{(x - 1)^{2}}$