Differentiation is a useful math tool that has many real-world uses, even for Year 9 students! Let’s look at some examples:

1. Physics & Motion: Differentiation helps us understand how things move. For example, if we want to know the position of a car described by the equation $s(t) = 5t^2$, we can use differentiation to find its speed. This gives us the velocity $v(t) = 10t$, which shows how the car's speed changes over time.

2. Economics: In business, differentiation can help figure out the best way to make money. If a company’s income is shown with the equation $R(x) = 100x - 5x^2$, we can differentiate to find out the best number of items ($x$) to sell for the highest profit.

3. Biology: Differentiation can also predict how fast a population grows. If we represent a population by the formula $P(t) = 50e^{0.1t}$, we can find the growth rate by using $P'(t) = 5e^{0.1t}$. This tells us how quickly the population is increasing.

These examples show how differentiation is all around us and helps us understand many different areas!