Differentiation

2026 Syllabus Objectives

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

  1. Understand the gradient of a curve at a point as the limit of the gradients of a suitable sequence of chords, and use the notations f'(x), f''(x), dy/dx, and d²y/dx² for first and second derivatives.

  2. Use the derivative of x^n (for any rational n), together with constant multiples, sums and differences of functions, and of composite functions using the chain rule.

  3. Apply differentiation to gradients, tangents and normals, increasing and decreasing functions and rates of change (including connected rates of change).

  4. Locate stationary points and determine their nature, and use information about stationary points in sketching graphs.

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1. Understanding Gradient as a Limit

What is a gradient?

The gradient of a straight line tells us how steep the line is. It measures how much the line goes up (or down) for every step we take to the right.

For a straight line, the gradient is the same everywhere. But for a curve, the steepness changes at different points along the curve.

The gradient of a curve at a point

To find the gradient of a curve at a specific point, we need to think about the tangent line at that point. A tangent is a straight line that just touches the curve at one point without crossing through it.

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

How do we find this gradient?

We use an idea involving chords. A chord is a straight line that joins two points on a curve.

Let's say we want to find the gradient at point A on a curve. We can:

  1. Pick another point B on the curve, close to A
  2. Draw a chord (straight line) from A to B
  3. Calculate the gradient of this chord

Now, here's the clever part: if we move point B closer and closer to point A, the chord gets closer and closer to being the tangent line.

As the distance between A and B gets smaller and smaller (we call this distance h), the gradient of the chord approaches the gradient of the tangent.

Example: Consider the curve y = x³ and the point where x = 2.

  • One point has coordinates (2, 8) because when x = 2, y = 2³ = 8
  • Another point close to it has coordinates (2 + h, (2 + h)³)
  • The gradient of the chord joining these points is: [(2 + h)³ - 8] / h

As h gets smaller and smaller (approaches zero), this gradient approaches the actual gradient of the curve at x = 2.

This is what we mean by a limit – the value that something approaches as we keep making a change.

Important: You don't need to do complicated calculations using limits. You just need to understand the basic idea that the gradient at a point is found by looking at what happens to chords as they get shorter and shorter.

Notation for derivatives

When we find the gradient function of a curve, we call it the derivative. There are different ways to write derivatives:

First derivative (the gradient function):

  • If y = f(x), we can write the derivative as:
    • f'(x) (read as "f prime of x" or "f dashed x")
    • dy/dx (read as "dee y by dee x")

Second derivative (the rate of change of the gradient):

  • This tells us how the gradient itself is changing
  • We can write it as:
    • f''(x) (read as "f double prime of x" or "f double dashed x")
    • d²y/dx² (read as "dee two y by dee x squared")

These are just different ways of writing the same thing. You'll see both notations used in questions and should be comfortable with both.

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