O Level differentiation chain rule worked examples Singapore

Quick answer

Singapore O Level Differentiation guide for students.

Many students find the chain rule in differentiation challenging because it requires careful application of nested functions. It's easy to get lost in the steps, especially under exam pressure. But fear not, with a structured approach, you can conquer these questions confidently.

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Understanding the Chain Rule

The chain rule is essential when differentiating composite functions. If you have a function 𝑦 = 𝑓(𝑔(𝑥)), the chain rule states that the derivative 𝑦' is $𝑓'(𝑔(𝑥)) \cdot 𝑔'(𝑥)$. This means you differentiate the outer function first, then multiply by the derivative of the inner function.

Step-by-Step Worked Examples

Example 1: Differentiating $𝑦 = (3𝑥^2 + 2𝑥)^5$

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Let's break this down:

1. Identify the outer and inner functions:
   Here, the outer function is $𝑢^5$ where $𝑢 = 3𝑥^2 + 2𝑥$.

2. Differentiate the outer function:
   The derivative of $𝑢^5$ with respect to 𝑢 is $5𝑢^4$.

3. Differentiate the inner function:
   The derivative of $3𝑥^2 + 2𝑥$ with respect to 𝑥 is 6𝑥 + 2.

4. Apply the chain rule:
   Combine these using the chain rule:
   $𝑦' = 5(3𝑥^2 + 2𝑥)^4 \cdot (6𝑥 + 2)$

5. Simplify the expression:
   $𝑦' = 5(3𝑥^2 + 2𝑥)^4 \cdot (6𝑥 + 2)$

Example 2: Differentiating $𝑦 = \sqrt{2𝑥^3 + 5𝑥}$

1. Rewrite the function:
   Express the square root as a power: $𝑦 = (2𝑥^3 + 5𝑥)^{1/2}$.

2. Identify the outer and inner functions:
   The outer function is $𝑢^{1/2}$ where $𝑢 = 2𝑥^3 + 5𝑥$.

3. Differentiate the outer function:
   The derivative of $𝑢^{1/2}$ is $\frac{1}{2}𝑢^{-1/2}$.

4. Differentiate the inner function:
   The derivative of $2𝑥^3 + 5𝑥$ is $6𝑥^2 + 5$.

5. Apply the chain rule:
   $𝑦' = \frac{1}{2}(2𝑥^3 + 5𝑥)^{-1/2} \cdot (6𝑥^2 + 5)$

6. Simplify the expression:
   $𝑦' = \frac{6𝑥^2 + 5}{2\sqrt{2𝑥^3 + 5𝑥}}$

Example 3: Differentiating $𝑦 = 𝑒^{4𝑥^2 + 3𝑥}$

1. Identify the outer and inner functions:
   The outer function is $𝑒^𝑢$ where $𝑢 = 4𝑥^2 + 3𝑥$.

2. Differentiate the outer function:
   The derivative of $𝑒^𝑢$ is $𝑒^𝑢$.

3. Differentiate the inner function:
   The derivative of $4𝑥^2 + 3𝑥$ is 8𝑥 + 3.

4. Apply the chain rule:
   $𝑦' = 𝑒^{4𝑥^2 + 3𝑥} \cdot (8𝑥 + 3)$

5. Simplify the expression:
   $𝑦' = (8𝑥 + 3)𝑒^{4𝑥^2 + 3𝑥}$

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Example 4: Differentiating $𝑦 = \ln(5𝑥^4 + 7)$

1. Identify the outer and inner functions:
   The outer function is $\ln(𝑢)$ where $𝑢 = 5𝑥^4 + 7$.

2. Differentiate the outer function:
   The derivative of $\ln(𝑢)$ is $\frac{1}{𝑢}$.

3. Differentiate the inner function:
   The derivative of $5𝑥^4 + 7$ is $20𝑥^3$.

4. Apply the chain rule:
   $𝑦' = \frac{1}{5𝑥^4 + 7} \cdot 20𝑥^3$

5. Simplify the expression:
   $𝑦' = \frac{20𝑥^3}{5𝑥^4 + 7}$

Common Mistakes Students Make

1. Forgetting to multiply by the derivative of the inner function.
2. Mixing up the order of differentiation.
3. Not simplifying expressions fully.

Exam Tip

In Singapore O Level exams, differentiation questions often test multiple techniques in a single problem. Practice combining the chain rule with the product and quotient rules to improve your flexibility. Always show each step clearly to maximise partial credit, even if your final answer is incorrect.

Practise with free question sets

Work through more exam-style questions with answers and step-by-step solutions:

• 35+ O Level Differentiation Practice Questions for 2026 (based on Singapore MOE Syllabus)
• Topic study hub

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