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Further Applications of Differentiation

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Further Applications of Differentiation

Edexcel IAL Maths: Pure 4

The Big Idea: Differentiation isn't just about finding gradients and optimising functions. In this chapter, you'll learn how to apply the chain rule to link multiple changing quantities together, and use derivatives to set up equations that describe real-world situations.

Chapter Overview

  • Master connected rates of change and the chain rule in complex contexts
  • Understand how to link multiple rates of change in real-world problems
  • Distinguish between variables (things that change) and constants (things that stay fixed)
  • Set up and solve connected rates problems systematically
  • Work with differential equations (equations containing derivatives)
  • Apply these skills to geometry problems (spheres, cones, cylinders, etc.)
  • Avoid common exam traps and mistakes

Topic 1: Connected Rates of Change

Using the chain rule to link multiple rates of change when more than two variables are involved in a problem.

What Are Connected Rates of Change?

Connected rates of change is about solving problems where multiple quantities are changing at the same time, and you need to find how fast one of them is changing by using information about another.

The key idea: If you have a situation with more than two variables, you can use the chain rule to create an equation that connects all the rates of change together.

A Real-World Example

Imagine a sphere is being inflated. As it expands:

  • The volume is increasing at a known rate (e.g., 10 cm³/sec)
  • You want to find how fast the surface area is increasing at a particular moment

You can't answer this directly because volume and surface area are related through a different variable (the radius). This is where the chain rule comes in—it lets you connect these two rates through the radius.

💡 Key Point: Connected rates problems are really just the chain rule in disguise. You're linking rates together to find the one you need.

Differential Equations

Equations that contain derivatives (rates of change) are called differential equations. In this chapter, you're learning to set up these equations from word problems. (Later, you'll learn to solve them using integration.)

The Chain Rule for Connected Rates

If you have variables that are all changing with respect to time, the chain rule lets you connect their rates of change.

The Chain Rule (Two Variables)
dA/dt = dA/ds × ds/dt
In words: The rate of change of A with respect to time equals the rate of change of A with respect to s, multiplied by the rate of change of s with respect to time.

How This Works

Think of it as a chain of connections:

You know: ds/dt (e.g., side length is changing at 3 cm/sec)

You can find: dA/ds by differentiating the formula for A in terms of s

You want: dA/dt (e.g., how fast the area is changing)

The chain rule gives you: dA/dt = dA/ds × ds/dt

🔗 Remember:

Three or More Variables

Variables vs Constants: This Matters!

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Also in the full note
  • How to Set Up a Connected Rates Problem
  • Worked Example 1: Expanding Cube
  • Worked Example 2: Expanding Sphere
  • Common Mistakes (And How to Avoid Them)
  • What to Memorise
  • Concepts Checklist
  • Exam Tips & Examiner Traps
  • Variables (Things That Change)
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More Pure Mathematics 4 topics

Proof by Contradiction
Proof by Contradiction
Partial Fractions
Partial Fractions with Linear Denominators
Integration by Substitution
General Binomial Expansion
General Binomial Expansion
Implicit Differentiation
Further Integration
Connected Rates of Change