Differentiate standard functions and use the product, quotient and chain rules to differentiate products, quotients and composite functions

What this dot point is asking

SEAB wants you to differentiate the standard functions and combine them using the product, quotient and chain rules, so you can differentiate any expression built from powers, exponentials, logarithms and trigonometric functions.

The answer

Standard derivatives

The library you must know:

The chain rule

For a composite $y = \mathrm{f}(\mathrm{g}(x))$:

Differentiate the outer function keeping the inner intact, then multiply by the derivative of the inner. This is the rule behind $\dfrac{d}{dx}e^{3x} = 3e^{3x}$.

The product rule

For $y = uv$:

The quotient rule

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

Note the order in the numerator (the derivative of the top times the bottom, minus top times derivative of bottom) and the squared denominator.

Combining the rules

Real expressions mix the rules: a quotient whose parts are themselves products, or a chain inside a product. Identify the outermost structure first, then work inward, applying the chain rule wherever a composite appears.

Nesting the rules: a worked structure

The hardest H2 differentiation questions nest one rule inside another, and the reliable approach is to name the outermost operation first. For $y = x^2 \tan(3x)$, the outermost structure is a product ($u = x^2$, $v = \tan 3x$), but differentiating $v$ needs the chain rule: $v' = 3\sec^2(3x)$. So $\tfrac{dy}{dx} = 2x\tan(3x) + x^2 \cdot 3\sec^2(3x)$. The discipline of asking "is the whole thing a product, a quotient, or a composite?" before differentiating any inner piece prevents the common error of applying the chain rule to the wrong layer. Work strictly from the outside in, and apply the chain rule at every composite you meet on the way down.

Logarithmic differentiation

When a function is a complicated product, quotient, or has a variable in both the base and the exponent (such as $y = x^x$), taking the natural log of both sides first turns products into sums and powers into coefficients, which are far easier to differentiate. For $y = x^x$, write $\ln y = x\ln x$, differentiate implicitly to get $\tfrac{1}{y}\tfrac{dy}{dx} = \ln x + 1$, then multiply by $y$: $\tfrac{dy}{dx} = x^x(\ln x + 1)$. Logarithmic differentiation is the technique of choice for $y = x^x$ and similar forms that none of the three basic rules handle directly, and it draws together the log laws with implicit differentiation.

Examples in context

Example 1. Rate of cooling. A temperature model $T = 20 + 60e^{-0.1t}$ has $\dfrac{dT}{dt} = -6e^{-0.1t}$ by the chain rule, giving the instantaneous cooling rate, which is steepest at the start and flattens as $t$ grows.

Example 2. Marginal revenue. If revenue is $R = p \cdot q(p)$ where quantity $q$ depends on price $p$, the product rule gives $\dfrac{dR}{dp} = q + p\dfrac{dq}{dp}$, the economist's marginal-revenue decomposition.

Try this

Q1. Differentiate $y = (3x + 1)^5$. [2 marks]

• Cue. Chain rule: $5(3x + 1)^4 \cdot 3 = 15(3x + 1)^4$.

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

• Cue. Product rule: $\cos x - x\sin x$.

Q3. Differentiate $y = \dfrac{e^x}{x}$. [2 marks]

• Cue. Quotient rule: $\dfrac{e^x x - e^x}{x^2} = \dfrac{e^x(x - 1)}{x^2}$.