14.1 Derivative Concept

2026 📋 Syllabus Objectives

By the end of this topic, you should be able to:

  • Understand the idea of a derived function — meaning you can explain what a derivative is and what it tells you about a curve.
  • Understand informally what a limit is — you do not need to perform differentiation from first principles or use formal limit calculations.

Main Content

https://cambridgeassistant.com

1. What Is a Curve's Gradient?

You already know how to find the gradient (steepness) of a straight line — it stays the same everywhere on the line.

But a curve is different. Its steepness changes from point to point. Think of a hill that gets steeper as you climb — the gradient at the bottom is different from the gradient near the top.

So the big question is: how do we find the gradient of a curve at any specific point?

The answer is: we use differentiation.

2. The Idea of a Limit (Informally)

Imagine you are standing on a curve at a point A. Now imagine a second point B very close to A on the same curve. You can draw a straight line through both A and B — this line is called a chord.

Now, imagine sliding B closer and closer to A — so close that the gap between them is almost nothing. As this happens, the chord gets closer and closer to a straight line that just touches the curve at A. That special line is called the tangent to the curve at A.

The gradient of that tangent line is the gradient of the curve at point A.

💡 Key idea: As the gap between A and B gets smaller and smaller (approaching zero), the gradient of the chord approaches a fixed value. That fixed value is called the limit, and it gives us the gradient of the curve at that point.

You do not need to calculate this using algebra step-by-step (that technique is called "first principles" and is outside your syllabus). You just need to understand the idea — that as two points get closer and closer together, the chord between them approaches the tangent, and the gradient approaches a specific value.

3. What Is a Derived Function?

When you differentiate a function (i.e. apply the rules of differentiation to it), you get a new function. This new function is called the derived function, or the derivative.

📌 In plain English: The derived function is a rule that tells you the gradient of the original curve at any point you choose.

For example:

  • The original function might be y = x²
  • After differentiating, the derived function is dy/dx = 2x
  • This means: at any point on the curve y = x², the gradient equals 2 times the x-value at that point

So if you want the gradient at x = 3, you substitute: gradient = 2(3) = 6 If you want the gradient at x = −1, you substitute: gradient = 2(−1) = −2

The derived function works like a gradient calculator for the original curve.

Sign in to view full notes