12.1PureCore

Gradients of Curves

A straight line has one fixed gradient, but a curve’s steepness changes from point to point. Differentiation is the tool for finding that changing gradient — and it all starts with the idea of a tangent.

25 min Video by Zeeshan Zamurred Differentiation

Edexcel Differentiation — full chapter playlist (Zeeshan Zamurred)Watch the full walkthrough before the notes below.

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What you'll be able to do

Understand that a curve’s gradient varies along it
Define the gradient at a point as the gradient of the tangent
See how chords approach the tangent
Interpret gradient as a rate of change

Gradient at a point

The gradient of a curve at a point is defined as the gradient of the $tangent$ — the straight line that just touches the curve there. Unlike a line, this value changes as you move along the curve.

Tip — On a curve, “the gradient” always means “the gradient at a particular point”.

Chords approaching the tangent

Take a point $P$ and a nearby point $Q$ on the curve. The line $P Q$ is a $chord$ , and its gradient approximates the curve’s gradient at $P$ . As $Q$ slides towards $P$ , the chord’s gradient gets closer and closer to the tangent’s gradient — the foundation of differentiation from first principles.

gradient of chord = \frac{change in y}{change in x} \to tangent as Q \to P

Shrink the gap to zero and the chord becomes the tangent.

Gradient as a rate of change

A gradient measures how fast $y$ changes as $x$ changes. This is why differentiation models real rates — speed (rate of change of distance), and many others throughout the course.

Formula recap

gradient at P = gradient of tangent at P

Definition.

chord gradient \to tangent gradient as Q \to P

First-principles idea.

gradient = rate of change of y with x

Interpretation.

Common mistakes to avoid

Treating a curve as if it has one single gradient.

A curve’s gradient changes at every point.

Confusing a chord with a tangent.

A chord joins two points; a tangent touches at one — chords approach the tangent.

Key takeaways

A curve’s gradient varies along it.
Gradient at a point = gradient of the tangent there.
Chord gradients approach the tangent as the points get closer.
Gradient represents a rate of change.

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