Year 12 Calculus: The product rule

The product rule is used when a function is written as one factor multiplied by another. Differentiate one factor at a time while keeping the other unchanged.

Use the pattern "first times derivative of second plus second times derivative of first". Writing the two factors separately before starting helps avoid missing a term.

Main rule

If $y = u (x) v (x)$ , then $d y / d x = u d v / d x + v d u / d x$ .

Worked example

Problem: Differentiate

y = x^{2} (x + 4)

Let $u = x^{2}$ and $v = x + 4$ .
Then $d u / d x = 2 x$ and $d v / d x = 1$ .
Apply the product rule: $d y / d x = x^{2} (1) + (x + 4) (2 x)$ .
Expand: $d y / d x = x^{2} + 2 x^{2} + 8 x$ .
Simplify: $d y / d x = 3 x^{2} + 8 x$ .

Answer:

d y / d x = 3 x^{2} + 8 x

When the product rule is needed

If a function is written as two clear factors multiplied together, the product rule is often the most direct method. Sometimes the function can be expanded first, which may be easier for simple polynomials. In harder expressions, though, the product rule keeps the structure visible and avoids unnecessary algebra.

Worked example 2

Problem: Differentiate

y = (2 x - 1) e^{x}

Let $u = 2 x - 1$ and $v = e^{x}$ .
Then $d u / d x = 2$ and $d v / d x = e^{x}$ .
Apply the product rule: $d y / d x = u d v / d x + v d u / d x$ .
Substitute: $d y / d x = (2 x - 1) e^{x} + e^{x} (2)$ .
Factor: $d y / d x = e^{x} (2 x + 1)$ .

Answer:

d y / d x = e^{x} (2 x + 1)

Common mistakes

Differentiating both factors at once and multiplying the derivatives together.
Losing one term of the sum in the product rule formula.
Expanding too early and creating avoidable algebraic errors.

The product rule matters because many practical functions are built from parts that each behave differently. One factor may be polynomial while the other is exponential, logarithmic, or trigonometric. Differentiating those functions term by term is not possible unless you first respect the multiplicative structure. Seeing that structure early is half the battle.

In revision, it helps to write the two factors on separate lines before differentiating. That forces you to keep track of which term stays fixed and which one changes. The rule itself is short, but most errors happen because the two factors were never identified clearly in the first place.

Revision focus

In calculus, one of the best habits is to decide what role the expression is playing before you touch a rule. Is it a product, a composite function, an implicitly defined curve, or an antiderivative problem? That classification step is not wasted time. It is the step that usually determines whether the next line of working will be short and accurate or long and unstable. Strong calculus work looks calm because the structure is identified early, not because the algebra is magically easier.

It is also worth building a habit of interpretation after the algebra is finished. A derivative is not just a formula. It represents a slope, a rate, or a turning condition. An integral is not just a reverse power rule. It may represent accumulated change or a signed area over an interval. If you say that interpretation out loud at the end of each question, the topic becomes much more coherent and your solutions become much easier to check.