Combining Differentiation Rules (Product, Quotient, Chain)
Learn combining differentiation rules for NSW Year 12 Mathematics Advanced (NESA outcome MAV-12-04). Many functions are built by multiplying, dividing or nesting simpler ones, such as \(x\) times \(e^x\) or \(\sin^2 x\), and each kind of build needs its own rule.
You will learn to choose and combine the product rule, quotient rule and chain rule for functions made from powers, exponentials, logarithms and trig functions, and set out working clearly — the fluency that makes curve sketching and optimisation manageable.
Theory
Combine the product, quotient and chain rules when functions are multiplied, divided or nested. This Year 12 Mathematics Advanced topic (MAV-12-04) applies all three rules.
Product: \(\dfrac{d}{dx}[uv]=u'v+uv'\). Quotient: \(\dfrac{d}{dx}\!\left[\dfrac{u}{v}\right]=\dfrac{u'v-uv'}{v^{2}}\). Chain: \(\dfrac{d}{dx}f(g(x))=f'(g(x))\,g'(x)\).
The first step is always to spot which structure you have.
Method
- Classify: product, quotient, or composite?
- Apply the matching rule, differentiating each part.
- Simplify the result.
| \(\dfrac{dy}{dx}\) | \(=\) | \((1)e^{x}+x e^{x}\) |
| \(=\) | \((1+x)e^{x}\) |
| \(\dfrac{dy}{dx}\) | \(=\) | \(\dfrac{(1)(x+1)-x(1)}{(x+1)^{2}}\) |
| \(=\) | \(\dfrac{1}{(x+1)^{2}}\) |
| \(\dfrac{dy}{dx}\) | \(=\) | \(5(2x+1)^{4}\cdot2\) |
| \(=\) | \(10(2x+1)^{4}\) |
| \(\dfrac{dy}{dx}\) | \(=\) | \(2x\sin x+x^{2}\cos x\) |
Common pitfalls
Frequently asked questions
What is the product rule?
The derivative of u times v is u prime v plus u v prime.
What is the quotient rule?
The derivative of u over v is (u prime v minus u v prime) all over v squared.
How do you know which differentiation rule to use?
Look at the structure: use the product rule for a product, the quotient rule for a fraction, and the chain rule for a function inside another function.
How do you differentiate x times e to the x?
By the product rule it is e to the x plus x e to the x, which factorises to (1 plus x) e to the x.