ELI5: how to work out growth or speed rates

To find a growth rate or speed rate in NCEA Level 2 Maths, you use a tool called calculus differentiation to find the gradient, which tells you how fast something is changing at an exact moment in time.

Imagine you are growing a sunflower. If you only measure it on Monday and Friday, you find the average rate of change, which is just the total height gained divided by the days passed. But on Wednesday, the sunflower might be growing much faster than on Friday. In Level 2 Maths, we want to know the exact speed at that very second.

Here is the simple recipe to find that speed:

• Step 1: Start with your equation. You will get a rule connecting time and amount. For example, the height of your sunflower might be $h = 3t^2$, where $t$ is the number of days.
• Step 2: Differentiate it. This is a maths trick called the power rule. You multiply the number in front by the power, and then reduce the power by one. So, the $2$ comes to the front to multiply with $3$ (making $6$), and the power drops to $1$. Your new gradient function is the rate: $\text{Rate} = 6t$.
• Step 3: Plug in the time. If you want to know how fast the sunflower is growing on day 4, replace $t$ with 4. The speed is $6 \times 4 = 24$ centimetres per day.

• Distance tells you where you are.
• Differentiating distance gives you velocity, which is the rate of speed.
• Differentiating velocity gives you acceleration, which is the rate at which your speed is growing.