Q.Differentiate : $\dfrac{x^2+x+1}{x^2-x+1}$.

Quotient rule: $\dfrac{u'v-uv'}{v^2}$ with $u=x^2+x+1,\ v=x^2-x+1$ gives $\dfrac{2(1-x^2)}{(x^2-x+1)^2}$.

A quotient-rule differentiation from the Differential Calculus unit of the Tamil Nadu HSC Class-11 Business Mathematics syllabus.

Step 1 — Set up. Let $u=x^2+x+1$ and $v=x^2-x+1$, so $u'=2x+1$ and $v'=2x-1$. The quotient rule gives

$\frac{dy}{dx}=\frac{u'v-uv'}{v^2}.$

Step 2 — Expand $u'v$.

$(2x+1)(x^2-x+1)=2x^3-x^2+x+1.$

Step 3 — Expand $uv'$.

$(x^2+x+1)(2x-1)=2x^3+x^2+x-1.$

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