# Rates of Change and the Derivative

> From average speed to the gradient of a curve. Limits, first principles and what differentiable means.

From HSC Maths Advanced on WhizzNotes, free study notes for NSW students: https://whizznotes.com/mathematics-advanced/rates-and-the-derivative/. Updated 3 November 2026.
Year 11 and Year 12 boxes go deeper than the Year 10 text around them.

### Story time: The speed camera
Hannah drives 120 km from Sydney to Newcastle in 1.5 hours. Her average speed is 80 km/h. But a speed camera on the freeway does not care about her average. It wants her speed at **one exact instant**, the moment she passes it.

How can you have a speed at an instant, when no time passes and no distance is covered? That puzzle took mathematicians centuries to solve. The answer is the **derivative**, the main idea of calculus.

**In short:** The derivative of a function is its rate of change, which is the gradient of the tangent to its graph at each point. It is written $f'(x)$ or $\dfrac{dy}{dx}$ and found from first principles as $f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$.

## Starting point: Years 9 and 10

### Rate of change

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

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

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

### Worked example: Average rate from a table
A plant is 4 cm tall on day 2 and 13 cm tall on day 5. The average growth rate is

$$\frac{13 - 4}{5 - 2} = 3 \text{ cm per day}$$

### Straight lines and curves

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.

*Diagram: The curve y equals x squared with a secant line through P and Q and a tangent line at P*
*Slide Q towards P and the secant turns into the tangent.*

### Key idea: The big idea
Take a second point Q on the curve. The line PQ (a **secant**) has a gradient you can calculate. Now slide Q closer and closer to P. The secant's gradient settles on one number. That number is the gradient of the curve at P.

### Worked example: Closing in on the tangent
For $y = x^2$ at $P(1, 1)$:

| Q | Gradient of PQ |
|---|---|
| $(2, 4)$ | $\dfrac{4 - 1}{2 - 1} = 3$ |
| $(1.1, 1.21)$ | $\dfrac{0.21}{0.1} = 2.1$ |
| $(1.01, 1.0201)$ | $\dfrac{0.0201}{0.01} = 2.01$ |

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

### Watch out: Average is not instantaneous
Average rate uses two points far apart. Instantaneous rate is the gradient of the tangent at one point. They are only equal for straight lines.

### Calculator tip: Check a gradient
If your calculator has a $\frac{d}{dx}$ key, it gives the gradient of a curve at a point, for example the derivative of $x^3$ at $x = 2$ is 12. Use it only to check: questions need the working. Use radian mode for trig functions.

**Mind map**

```text
The derivative
  Rates
    Change in y over change in x
    Average rate: secant
    Instant rate: tangent
  Limits
    Let h shrink to 0
  First principles
    f'(x) = lim (f(x+h) - f(x))/h
  Notation
    f'(x)
    dy/dx
  Differentiable
    Smooth and unbroken
    No corners or gaps
```

### Year 11: Limits and first principles
Let P be at $x$ and Q be at $x + h$. The secant gradient is

$$\frac{f(x + h) - f(x)}{h}$$

Now let $h$ get closer and closer to 0. This is called taking a **limit**. The result is the **derivative** $f'(x)$:

$$f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$$

**Example.** Differentiate $f(x) = x^2$ from first principles.

$$\frac{(x + h)^2 - x^2}{h} = \frac{2xh + h^2}{h} = 2x + h$$

As $h \to 0$, this becomes $2x$. So $f'(x) = 2x$. At $x = 1$ the gradient is 2, matching the table.

**Notation.** If $y = f(x)$, the derivative can be written $f'(x)$ or $\dfrac{dy}{dx}$ (read "dee y by dee x"). It is the gradient function: put in any $x$, get the gradient there.

**Another way to write it.** Letting a second point $x$ slide towards $a$ gives the same derivative:
$$f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}$$

**What a derivative means.** $f'(a)$ is a rate, with units of "$y$ units per $x$ unit". If $H(t)$ is a tree's height in metres after $t$ years, $H'(4) = 0.5$ means that at 4 years the tree is growing at 0.5 m per year at that moment.

**Estimating nearby values.** Close to $x = a$, the curve is almost its tangent, so $f(a + h) \approx f(a) + h f'(a)$. If $f(3) = 10$ and $f'(3) = 2$, then $f(3.1) \approx 10 + 0.1 \times 2 = 10.2$.

### Year 11: Differentiability
A function is **differentiable** at a point if it has a single clear tangent there.

- $y = |x|$ is not differentiable at $x = 0$. It has a sharp corner, and the gradients on each side are $-1$ and $1$.
- A function with a jump or a gap is not differentiable at the break.
- A curve with a vertical tangent, like $y = \sqrt[3]{x}$ at $x = 0$, is not differentiable there.

**Smooth and unbroken** means differentiable.

**Joining two rules smoothly.** For a piecewise function to be differentiable at the join, the two pieces must meet (same value) **and** have the same gradient there. For $f(x) = x^2$ when $x \leq 2$ and $f(x) = ax + b$ when $x > 2$: the gradient of $x^2$ at 2 is 4, so $a = 4$; the values match when $4 = 2a + b$, so $b = -4$.

### Year 12: Continuity and the derivative as a function
A function is **continuous** at $x = a$ if you can draw it through $a$ without lifting the pen: $\displaystyle\lim_{x \to a} f(x) = f(a)$. Every differentiable function is continuous, but not every continuous function is differentiable (think of the corner in $|x|$).

The derivative $f'(x)$ is itself a function, so it has a graph. Where $f$ rises, $f'$ is above the x-axis. Where $f$ has a turning point, $f'$ crosses zero. Sketching $f'$ from $f$ is a common exam skill.

## Quick summary

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

## Quiz

Try these questions from the chapter quiz.

1. A car travels 150 km in 2 hours. Its average speed is:
   - A. 300 km/h
   - B. 75 km/h
   - C. 50 km/h
   - D. 152 km/h
   - Answer: B. 75 km/h. $150 \div 2 = 75$ km/h.

2. Water in a tank rises from 20 L at 1 pm to 80 L at 4 pm. The average rate is:
   - A. 60 L/h
   - B. 20 L/h
   - C. 30 L/h
   - D. 80 L/h
   - Answer: B. 20 L/h. $\dfrac{80 - 20}{4 - 1} = 20$ L per hour.

3. The gradient of a straight line:
   - A. is the same everywhere
   - B. changes at every point
   - C. is always positive
   - D. is always zero
   - Answer: A. is the same everywhere. A line has constant steepness.

The whole chapter quiz is on the page, for signed-in students: https://whizznotes.com/mathematics-advanced/rates-and-the-derivative/#quiz