Introduction to Derivatives · Grade 12

When do you use the product, quotient and chain rules?

Look at how the function is built. Two things multiplied need the product rule, one thing divided by another needs the quotient rule, and a function sitting inside another function needs the chain rule. Identifying the structure correctly is most of the work; the formulas themselves are short.

Learning objectives

Recognising which it is

Ask how the expression was assembled. Were two separate functions multiplied together? Was one divided by another? Or was one substituted into another?

That single question decides the rule, and answering it out loud before writing anything saves more marks than knowing the formulas faster. Most errors in this topic are structural, not algebraic.

A useful test for the chain rule: if you would need brackets to describe the inside, there is an inside. A bracket raised to a power, or a root over a whole expression, is a composition.

The three formulas

Product: the derivative of the first times the second, plus the first times the derivative of the second. Both terms appear, and dropping one is the usual mistake.

Quotient: the derivative of the top times the bottom, minus the top times the derivative of the bottom, all over the bottom squared. The order in the numerator matters because of the minus sign.

Chain: the derivative of the outer function evaluated at the inner one, multiplied by the derivative of the inner. That final multiplication is the part people forget.

When more than one rule applies

Work from the outside in. Identify the outermost structure first — is the whole thing a product, a quotient, or a composition — and apply that rule, then deal with the pieces.

Nested rules are common at this level and are not harder, only longer. Writing each piece and its derivative on a separate line before assembling keeps them manageable.

Worked examples

  1. Differentiate f(x) = x squared times (x + 3)

    1. This is a product of two functions
    2. Derivative of the first times the second: 2x times (x + 3)
    3. Plus the first times the derivative of the second: x squared times 1
    4. Expand and collect: 2x squared plus 6x plus x squared

    Answer: 3x squared plus 6x

  2. Differentiate f(x) = (2x + 1) to the fifth power

    1. A bracket raised to a power, so a composition — the chain rule
    2. Differentiate the outside: 5 times the bracket to the fourth
    3. Multiply by the derivative of the inside, which is 2

    Answer: 10 times (2x + 1) to the fourth power

  3. Differentiate f(x) = x over (x + 1)

    1. A quotient, so the quotient rule
    2. Top of the numerator: 1 times (x + 1) minus x times 1
    3. Simplify: x plus 1 minus x, which is 1
    4. Divide by the bottom squared

    Answer: 1 over (x + 1) squared

Common mistakes

Differentiating a product by differentiating each factor
The derivative of a product is not the product of the derivatives. Check on x times x: the product rule gives 2x, while multiplying the derivatives gives 1.
Forgetting the inner derivative in the chain rule
The chain rule ends with a multiplication by the derivative of the inside. Omitting it is invisible when the inside is just x, which is exactly why the habit fails when the inside is anything else.
Reversing the numerator in the quotient rule
The minus sign makes the order matter. Swapping the two terms gives the derivative with the wrong sign, and everything built on it — slopes, extrema — comes out backwards.

What to remember

More in Introduction to Derivatives