Q.Differentiate $y = \dfrac{x+1}{x-1}$ with respect to $x$, using the quotient rule.

This is a ratio of two linear expressions, so the quotient rule applies directly, with $u=x+1$ as the numerator and $v=x-1$ as the denominator.

Step 1 — Identify $u$ and $v$. $u=x+1$, $v=x-1$ ($v\neq0$, i.e. $x\neq1$), so $y=\dfrac{u}{v}$.

Step 2 — Differentiate each part. $u'=1$, $v'=1$.

Step 3 — Apply the quotient rule.

Step 4 — Simplify the numerator. $(x-1)-(x+1) = x-1-x-1 = -2$. So

$\frac{dy}{dx} = \frac{-2}{(x-1)^2}$ …

Reversing the order of subtraction in the numerator (writing $uv'-u'v$ instead of $u'v-uv'$) silently flips the sign of the final answer — the order in the quotient-rule numerator must always be (derivative of numerator)×(denominator) minus (numerator)× …