15. Rates of Change and the Derivative

Starting point: Years 9 and 10

A rate compares how fast one quantity changes compared with another. Speed is distance per time. Cost per kilogram is money per mass.

$average rate of change = \frac{change in y}{change in x}$

On a graph, this is just the gradient of the straight line joining two points.

A straight line has the same gradient everywhere. A curve's steepness keeps changing. So for a curve we ask: how steep is it at one point?

The answer is the gradient of the tangent: the line that just touches the curve at that point and runs in the same direction.

Slide Q towards P and the secant turns into the tangent.

Worked exampleClosing in on the tangent

For $y = x^{2}$ at $P (1, 1)$ :

The gradients approach 2. The gradient of $y = x^{2}$ at $x = 1$ is 2.

The derivative

Rates

Limits

First principles

Notation

Differentiable

Average rate of change is the gradient of a secant: $\frac{Δ y}{Δ x}$ .
The gradient of a curve at a point is the gradient of the tangent there.
First principles: $f^{'} (x) = h \to 0 lim \frac{f ( x + h ) - f ( x )}{h}$ .
$f^{'} (x)$ and $\frac{d y}{d x}$ are the gradient function.
Differentiable means smooth and unbroken: no corners, jumps or vertical tangents.

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