§-Module

Topic 1: Further Differentiation and Applications

5 dot points across 5 inquiry questions. Click any dot point for a focused answer with worked past exam questions where available.

How do we use the first and second derivatives to sketch an accurate graph of a function?

Curve sketching combines intercepts, stationary points, concavity and end behaviour to produce an accurate graph.

A systematic method for sketching curves using intercepts, stationary points from the first derivative, and concavity and inflections from the second derivative.

8 min answer →

How do we use calculus to find the maximum or minimum value of a real-world quantity?

Optimisation problems use the first derivative to locate the maximum or minimum value of a quantity subject to a constraint.

A step-by-step method for solving optimisation problems: build the function, use a constraint to reduce to one variable, differentiate, solve for stationary points, and justify the maximum or minimum.

8 min answer →

How do we differentiate functions that are products or quotients of simpler functions?

The product and quotient rules differentiate functions formed by multiplying or dividing two differentiable functions.

How to use the product rule and quotient rule to differentiate functions built by multiplying or dividing two functions, with full worked SACE-style examples.

8 min answer →

What does the second derivative tell us about the shape of a curve?

The second derivative measures the rate of change of the gradient and determines concavity and points of inflection.

How the second derivative determines whether a curve is concave up or down, locates points of inflection, and classifies stationary points.

8 min answer →

How do we differentiate a function that is composed of one function inside another?

The chain rule differentiates composite functions by multiplying the derivative of the outer function by the derivative of the inner function.

How to differentiate composite functions with the chain rule, including the outside-times-inside method and how it combines with the product and quotient rules, with worked SACE-style examples.

8 min answer →