My example is from the real life situation of war. From experiments in physics we know that the acceleration due to gravity of a particle near the Earth's surface is about $-9.8 \frac{m}{s^2}$.
If you're an artilleryman in an army, you want your artillery shells to hit the enemy, or else theirs may hit you and kill you (game over). So you need to know how to angle your artillerygun and which direction to point it in so that when the shell lands, it blows up your enemy (rather than missing).
All you know is that you can control the direction your cannon points and the angle you fire it in in the air -- after it's fired gravity takes over and that $-9.8 \frac{m}{s^2}$ takes over the situation.
Calculus shows us that if $x(t)$ is the function representing the position of the artillery shell at time $t$, then its first derivative is the shell's velocity and its second derivative is the shell's acceleration. We know acceleration from physics (it's that $-9.8 \frac{m}{s^2}$ we had earlier)! We write this as $x''(t) = -9.8 \frac{m}{s^2}$.
From this equation (involving derivatives) you can calculate (using "integration") the position function $x(t)$ of the particle given the direction you fire it in and the angle you fire it at.
Why do you care about that? Because knowing $x(t)$ will tell you where your shell lands and thus whether your shot will kill the enemy or not. So you can do a quick calculation to determine which direction and which angle to fire in to ensure that your shell hits your target. The side that does this computation first and gets the shell in the air first will kill the other side, helping win the war.