Apply the product, quotient and chain rules to differentiate functions, and use the derivative in optimisation, rates of change and curve sketching

What this dot point is asking

SCSA Unit 3 extends your differentiation toolkit so you can handle functions built from several pieces multiplied, divided or nested inside one another, and then apply the derivative to real contexts: maxima and minima, rates of change, and sketching. This dot point is examined in both the calculator-free and calculator-assumed sections of the WACE written examination.

The three combining rules

The skill SCSA tests is recognising which rule (often several) a function needs and applying them in the correct order. Always differentiate the outermost structure first, then work inwards.

Standard derivatives you must know

• $\dfrac{d}{dx}(x^n) = n x^{n-1}$
• $\dfrac{d}{dx}(\sin x) = \cos x$, $\dfrac{d}{dx}(\cos x) = -\sin x$
• $\dfrac{d}{dx}(e^{x}) = e^{x}$, $\dfrac{d}{dx}(\ln x) = \dfrac{1}{x}$

Combining the rules

A single function can require all three rules. The strategy is to label the structure clearly before differentiating.

Rates of change

The derivative $\dfrac{dy}{dx}$ is the instantaneous rate of change of $y$ with respect to $x$. If a quantity $Q$ depends on time $t$, then $\dfrac{dQ}{dt}$ is the rate of change of $Q$. For example, if the volume of water in a tank is $V(t) = 50t - t^2$ litres, the inflow rate at time $t$ is $V'(t) = 50 - 2t$ litres per unit time, which is zero at $t = 25$ (the tank stops filling).

Optimisation

To find a maximum or minimum of a quantity in context:

1. Write the quantity to be optimised as a function of one variable, using any constraint to eliminate other variables.
2. Differentiate and solve $f'(x) = 0$ for the stationary points.
3. Classify each stationary point (sign of $f'$ either side, or the sign of $f''$).
4. Check the endpoints of the domain and interpret the answer in context.

The second derivative and curve sketching

The second derivative $f''(x)$ measures concavity:

• $f''(x) > 0$: the curve is concave up (holds water).
• $f''(x) < 0$: the curve is concave down.
• A point of inflection occurs where $f''$ changes sign.

The second derivative test classifies a stationary point $x = a$ where $f'(a) = 0$: if $f''(a) > 0$ it is a local minimum, and if $f''(a) < 0$ it is a local maximum. When $f''(a) = 0$ the test is inconclusive and you must use a sign table for $f'$.

A complete curve sketch combines intercepts, stationary points, the nature of each stationary point, points of inflection, and end behaviour.