Apply the product, quotient and chain rules, individually and in combination, to differentiate products, quotients and composite functions

What this dot point is asking

SEAB wants you to differentiate functions built by multiplying, dividing or composing simpler functions, using the product, quotient and chain rules. The skill is to recognise the structure of an expression, choose the right rule (or combination), and apply it accurately, especially the chain-rule factor that beginners forget.

The answer

The chain rule

For a composite function $y = f(g(x))$, differentiate the outer function keeping the inner intact, then multiply by the derivative of the inner:

This is why $\dfrac{d}{dx}(3x - 1)^5 = 5(3x - 1)^4 \times 3$; the extra factor $3$ is the derivative of the inside.

The product rule

For a product $y = uv$:

Differentiate each factor in turn, multiply by the other, and add.

The quotient rule

For a quotient $y = \dfrac{u}{v}$:

The numerator is the derivative of the top times the bottom, minus the top times the derivative of the bottom; the denominator is squared. The order of subtraction matters.

Combining the rules

Real functions mix the rules: a product whose factor is a composite, or a quotient whose parts are products. Identify the outermost structure first (is the whole thing a product, a quotient, or a composite?), then apply the chain rule wherever a composite appears inside.

Choosing which rule

A quick test of structure decides the rule. If the function is two things multiplied, use the product rule; if it is one thing divided by another, use the quotient rule; if it is a function inside another function (a power of a bracket, an exponential of an expression), use the chain rule. Many expressions need more than one rule, applied from the outside in.

Why the chain rule has an extra factor

The chain rule's inner-derivative factor is what differentiation contributes when the inside is not simply $x$. Forgetting it is the single most common differentiation error, so whenever you differentiate a bracket raised to a power, an exponential, or a trigonometric function of anything other than $x$, multiply by the derivative of the inside.

Examples in context

Example 1. Decaying oscillation. A damped vibration $y = e^{-t}\cos t$ is a product of a decaying exponential and a cosine; differentiating it with the product and chain rules gives the velocity of the oscillation at any instant.

Example 2. Concentration over time. A drug concentration modelled by a quotient such as $C = \dfrac{t}{t + 1}$ has its rate of change found by the quotient rule, telling a pharmacologist how quickly the level is rising or falling.

Try this

Q1. Differentiate $y = (x^2 + 1)^4$. [2 marks]

• Cue. Chain rule: $4(x^2 + 1)^3 \times 2x = 8x(x^2 + 1)^3$.

Q2. Differentiate $y = x\ln x$. [2 marks]

• Cue. Product rule: $\ln x + x\cdot\dfrac{1}{x} = \ln x + 1$.

Q3. Differentiate $y = \dfrac{\sin x}{x}$. [3 marks]

• Cue. Quotient rule: $\dfrac{x\cos x - \sin x}{x^2}$.