This is Chapter 14 – Introduction to Differential Calculus for the Year 11 Mathematical Methods course.

This chapter introduces differential calculus through the idea of rate of change. Students move from measuring how a function changes across an interval to asking how quickly it is changing at one particular point, creating the conceptual foundation for the derivative.

We begin with average rates of change, linking the calculation to the gradient of a secant line between two points on a graph. Students then investigate what happens as those points move closer together, developing an intuitive understanding of instantaneous rate of change.

The gradient of the tangent provides a geometric interpretation of this instantaneous change. Students estimate and calculate tangent gradients and connect them with real contexts such as velocity, growth and changing physical quantities. The distinction between average and instantaneous change is central: two functions can have the same overall change across an interval while behaving very differently at individual points.

The derivative function then brings these individual gradients together. Rather than finding the gradient at only one point, students begin to view the derivative as a new function that describes how the gradient of the original function changes across its domain.

These ideas form the basis of the more procedural differentiation techniques introduced in the following chapter and developed extensively in Year 12 Mathematical Methods. Calculus is used throughout physics, engineering, economics, medicine and computer science wherever rates of change, motion or optimisation need to be analysed.

Subchapters:

• 14A: Rates of change
  
• 14B: Instantaneous rates of change
• 14C: Finding the gradient of the tangent
• 14D: The derivative function

By the end of this chapter, students will be able to interpret average and instantaneous rates of change, connect derivatives with tangent gradients and understand the derivative as a function.