---
title: "O Level Additional Mathematics: Product Rule Differentiation Worked Examples"
excerpt: "Master the product rule for O Level AMath with clear, step-by-step worked examples."
category: "O Levels"
seoCluster: "o-level-amath-differentiation"
pageIntent: "worked-examples"
level: "O Level"
subject: "Additional Mathematics"
topic: "Differentiation"
thumbnail: ""
author:
name: "[Tutorly.sg](https://tutorly.sg/app)"
---
Understanding the product rule in differentiation can be a challenge for many O Level Additional Mathematics students. It's common to mix up the product rule with other differentiation techniques, leading to errors in exams. Let's break down this topic with worked examples and clear, step-by-step solutions to help you gain confidence and accuracy.
> “Stuck on a question? See simple explanations that help you understand fast.”
> [👉 Give it a try and turn confusion into clarity in minutes.](https://tutorly.sg/app)
## Understanding the Product Rule
The product rule is used when differentiating expressions that are products of two functions. If you have a function $y = u(x) \cdot v(x)$, the product rule states that:
$$\frac{dy}{dx} = u'(x) \cdot v(x) + u(x) \cdot v'(x)$$
Here, $u'(x)$ and $v'(x)$ are the derivatives of $u(x)$ and $v(x)$, respectively. Let's see how this works in practice with some examples.
## Worked Example 1: Simple Polynomial Product
**Problem:** Differentiate $y = (2 x^3)(x^2 + 1)$.
> “Access more than 1000+ past year papers to practice”
> [👉 Start a paper today and test yourself like it’s the real exam.](https://tutorly.sg/app)
**Solution:**
1. Identify the functions: $u(x) = 2 x^3$ and $v(x) = x^2 + 1$.
2. Find the derivatives: $u'(x) = 6 x^2$ and $v'(x) = 2 x$.
3. Apply the product rule:
$$\frac{dy}{dx} = (6 x^2)(x^2 + 1) + (2 x^3)(2 x)$$
4. Simplify each term:
$$= 6 x^4 + 6 x^2 + 4 x^4$$
5. Combine like terms:
$$\frac{dy}{dx} = 10 x^4 + 6 x^2$$
By identifying $u(x)$ and $v(x)$ and using their derivatives, we applied the product rule and simplified the expression.
## Worked Example 2: Trigonometric Functions
**Problem:** Differentiate $y = x \sin(x)$.
**Solution:**
1. Identify the functions: $u(x) = x$ and $v(x) = \sin(x)$.
2. Find the derivatives: $u'(x) = 1$ and $v'(x) = \cos(x)$.
3. Apply the product rule:
$$\frac{dy}{dx} = (1)(\sin(x)) + (x)(\cos(x))$$
4. Simplify the expression:
$$= \sin(x) + x \cos(x)$$
This example shows the application of the product rule to a polynomial and a trigonometric function, emphasizing the importance of correctly identifying $u(x)$ and $v(x)$.
## Worked Example 3: Exponential and Logarithmic Functions
**Problem:** Differentiate $y = (e^x)(\ln(x))$.
**Solution:**
1. Identify the functions: $u(x) = e^x$ and $v(x) = \ln(x)$.
2. Find the derivatives: $u'(x) = e^x$ and $v'(x) = \frac{1}{x}$.
3. Apply the product rule:
$$\frac{dy}{dx} = (e^x)(\ln(x))' + (e^x)'(\ln(x))$$
$$= (e^x)\left(\frac{1}{x}\right) + (e^x)(\ln(x))$$
4. Simplify the expression:
$$= \frac{e^x}{x} + e^x \ln(x)$$
> “Doing Secondary Science? Pick a topic and practise like it’s a real exam — with clear answers right after.”
> [👉 Try Tutorly now and start a Science topic in seconds.](https://tutorly.sg/app)
## Worked Example 4: Combined Functions
**Problem:** Differentiate $y = (x^2 + 3 x)(\cos(x))$.
**Solution:**
1. Identify the functions: $u(x) = x^2 + 3 x$ and $v(x) = \cos(x)$.
2. Find the derivatives: $u'(x) = 2 x + 3$ and $v'(x) = -\sin(x)$.
3. Apply the product rule:
$$\frac{dy}{dx} = (2 x + 3)(\cos(x)) + (x^2 + 3 x)(-\sin(x))$$
4. Simplify each term:
$$= (2 x + 3)\cos(x) - (x^2 + 3 x)\sin(x)$$
5. Rearrange into a cleaner form:
$$= 2 x \cos(x) + 3\cos(x) - x^2 \sin(x) - 3 x \sin(x)$$
## Common Mistakes Students Make
1. **Incorrect Function Identification:** Mixing up $u(x)$ and $v(x)$ leads to wrong derivatives.
2. **Forgetting to Apply Product Rule:** Often, students mistakenly apply the basic power rule instead.
3. **Simplification Errors:** Incorrect algebraic simplification can lead to final answer mistakes.
## Exam Tip
In Singapore O Level exams, product rule questions often combine with other differentiation rules such as the chain rule or quotient rule. Ensure you practice integrating these rules within a single problem. Clear working steps and proper notation are crucial for gaining full marks.
## Related Topics You Should Learn Next
- [O Level AMath Differentiation Guide](https://tutorly.sg/blog/o-level-amath-differentiation-guide)
- [O Level Differentiation Chain Rule Worked Examples Singapore](/blog/o-level-differentiation-chain-rule-worked-examples-singapore)
- [O Level Product Rule and Quotient Rule Differentiation Singapore](/blog/o-level-product-rule-and-quotient-rule-differentiation-singapore)
- [O Level Amath Differentiation Questions Singapore: A Complete Worksheet Practice Guide](/blog/o-level-amath-differentiation-questions-singapore)
- [O Level Additional Math Differentiation Complete Guide Singapore](/blog/o-level-additional-math-differentiation-complete-guide-singapore)
- [O Level Amath Differentiation](https://tutorly.sg/learn/o-level-amath-differentiation)
[Try practice on Tutorly](https://tutorly.sg/app)
“Practice PSLE Science questions and get clear, step-by-step answers instantly.”
👉 Try a question now and see how fast you can improve.
