14.2 Derivative Notation

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Main Content

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1. What Is Differentiation?

When you have a curve (like y = x²), the gradient (steepness) of the curve is different at every single point. Differentiation is the process of finding a rule — called the gradient function — that tells you the gradient of the curve at any point you choose.

The result of differentiation is called the derivative.

There are two common ways to write derivatives: Leibniz notation (using dy/dx) and function notation (using f'(x)). Both mean exactly the same thing — they just look different.

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2. Understanding δx and δy

Before we can understand derivative notation fully, we need to understand what the symbols δx and δy mean.

• The Greek letter δ (delta) means "a small change in".
• So δx means "a small change in x".
• And δy means "a small change in y".

Imagine this picture: You are standing on a curve at point A. You move a tiny step along the curve to reach point B, which is very close to A.

• The horizontal distance you moved is δx (a small increase in x).
• The vertical distance you moved is δy (a small increase in y).

The gradient of the straight line (called a chord) joining A to B is:

$\frac{\delta y}{\delta x}$

This is just rise over run — the same gradient formula you already know from straight lines.

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3. What Does δx → 0 Mean?

Now here is the key idea of differentiation.

If point B is far from A, the chord AB is a rough approximation of the curve's gradient at A. But if you move B closer and closer to A — making δx smaller and smaller — the chord gets closer and closer to the tangent (the line that just touches the curve at A).

When δx becomes infinitely small (essentially zero), the gradient of the chord becomes the exact gradient of the curve at point A.

We write this as:

$\delta x \to 0$

This is read as "δx tends to (approaches) zero" or "δx gets closer and closer to zero".

So the exact gradient of the curve at any point is what we get when δx → 0.

In plain English: We imagine shrinking the gap between A and B until B is practically sitting on top of A. At that moment, the chord becomes a tangent, and we get the true gradient of the curve.