Derivatives are really important for understanding how fast something is moving at a certain moment. This is called instantaneous velocity. Instantaneous velocity is about the speed of an object right now, not how fast it went on average over a longer time.

1. What is a Derivative?

A derivative shows how much a function is changing. If we have a function called $f(t)$, the derivative at a specific point $t = a$ can be calculated like this: $f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}$

2. How Does This Relate to Velocity?

When we think about where something is located over time, we call that position $s(t)$. The instantaneous velocity at a specific time $t$ is just the derivative of the position function: $v(t) = s'(t)$

3. A Simple Example:

Imagine a car's position is described by the equation $s(t) = 5t^2$. To find the instantaneous velocity, we take the derivative, which gives us: $v(t) = s'(t) = 10t$ So, if we look at the car's speed at $t = 3$ seconds, we calculate: $v(3) = 10(3) = 30 \text{ m/s}$ This means the car is going 30 meters per second at that moment.

4. In Summary:

Derivatives are a handy way to figure out instantaneous velocity. They are crucial for understanding motion in calculus.