This is Chapter 1 – Differential Calculus for the Year 12 Mathematical Methods course.

Differential calculus is one of the central ideas in Stage 2 Mathematical Methods. It provides a way of measuring instantaneous change and allows students to analyse how a function behaves at a particular point. The differentiation techniques developed in this chapter are used repeatedly throughout the rest of the course, so accuracy and confidence here are essential.

We begin with differentiation from first principles. Although later questions are usually completed using quicker rules, first principles explains where the derivative comes from and builds a genuine understanding of its meaning as a gradient and instantaneous rate of change. Students then develop the standard rules needed to differentiate powers and combinations of functions efficiently.

The chain, product and quotient rules allow increasingly complicated expressions to be handled. One of the main challenges is recognising the underlying structure of a function and selecting the correct rule before beginning the calculation. Students then apply these techniques to exponential, logarithmic and trigonometric functions, which are common throughout both school assessments and the final examination.

Second derivatives extend the analysis further by describing how the gradient itself is changing. This idea later supports the study of concavity, points of inflection, stationary points and curve shape.

Success in this chapter requires more than memorising formulas. Students must maintain strong algebraic control, simplify expressions carefully and present working in a clear sequence. Differential calculus is used extensively in physics, engineering, economics, medicine, computer science and any field concerned with rates of change, optimisation or mathematical modelling.

Subchapters:

• 1A: First principles

• 1B: Simple rules of differentiation
• 1C: The chain rule
• 1D: The product rule
• 1E: The quotient rule
• 1F: Derivatives of exponential functions
• 1G: Derivatives of logarithmic functions
• 1H: Derivatives of trigonometric functions
• 1I: Second derivatives

By the end of this chapter, students will be able to differentiate a broad range of functions and interpret first and second derivatives as meaningful descriptions of function behaviour.