Derivatives are really important for understanding how things move in the real world. They help us explain ideas like speed and how speed changes.

1. Velocity:

   • When we look at the position of something over time, we call it a position function, written as $s(t)$.
   • The first derivative of this function gives us the velocity function, $v(t)$:
     $v(t) = s'(t)$
   • This tells us how fast an object is moving and in which direction.

2. Acceleration:

   • The second derivative of the position function gives us the acceleration function, $a(t)$. This is how we see changes in velocity:
     $a(t) = v'(t) = s''(t)$
   • Acceleration shows us how the speed of an object is changing over time.

3. Tangent Lines:

   • We use derivatives to find tangent lines at any spot on a graph. This helps us predict where something will go based on how it’s moving right now.

By using these ideas, we can study and guess how moving things act. This applies to all sorts of situations, from cars on the road to planets in space.