Differentiation Chain Rule – O Level 2024 Qn

Introduction

Square roots often scare students, but they’re actually straightforward with the differentiation chain rule. In this question, you just rewrite the square root as a power, apply the chain rule, and simplify carefully—exactly the kind of method O Level markers love.

The Question / Scenario Explanation

Source: Differentiation O Level 2024 Paper 1

Find \(\frac{d}{dx}\sqrt{x^2 + x + 1}\).

Step-by-Step Solution / Explanation

Step 1: Rewrite the square root using indices

\(\sqrt{x^2 + x + 1} = (x^2 + x + 1)^{\tfrac{1}{2}}\)

Step 2: Apply the differentiation chain rule

Let \(u = x^2 + x + 1\).
Then \(\frac{d}{dx}(u^{\tfrac{1}{2}}) = \tfrac{1}{2}u^{-\tfrac{1}{2}} \cdot \frac{du}{dx}\).

So,
\(\frac{d}{dx}(x^2 + x + 1)^{\tfrac{1}{2}} = \tfrac{1}{2}(x^2 + x + 1)^{-\tfrac{1}{2}} \cdot (2x + 1)\).

Step 3: Write the final answer neatly in square root form

\(\frac{2x + 1}{2\sqrt{x^2 + x + 1}}\)

✅ Final Answer: \(\frac{2x + 1}{2\sqrt{x^2 + x + 1}}\)

Key Concepts Students Must Know

• Differentiation chain rule: differentiate the outside first, then multiply by the derivative of the inside.
• Rewriting \(\sqrt{u}\) as \(u^{\tfrac{1}{2}}\) makes differentiation easier.
• Power rule reminder: \(\frac{d}{dx}(u^n) = n u^{n-1} \cdot \frac{du}{dx}\).

Exam Tips / Common Mistakes

Exam Tips

• Always rewrite square roots as powers first.
• Keep the answer in a clean fraction form: numerator over \(2\sqrt{\cdot}\).
• Don’t expand \(x^2 + x + 1\)—it’s unnecessary and wastes time.

Common Mistakes

• Forgetting to multiply by \(\frac{d}{dx}(x^2 + x + 1) = 2x + 1\).
• Writing only \(\tfrac{1}{2}(x^2 + x + 1)^{-\tfrac{1}{2}}\) and forgetting the \((2x + 1)\).
• Incorrect simplification of negative indices: \(u^{-\tfrac{1}{2}} = \frac{1}{\sqrt{u}}\).

Parent Insight

Many students lose marks in differentiation not because they don’t know the topic, but because they skip steps under pressure. A consistent “rewrite → chain rule → simplify” routine helps them score reliably in O Level Paper 1 differentiation questions.

Conclusion

This differentiation chain rule question is solved by rewriting the square root as a power and applying the chain rule carefully. The derivative of \(\sqrt{x^2 + x + 1}\) is \(\frac{2x + 1}{2\sqrt{x^2 + x + 1}}\).

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