What is a derivative?
A derivative is the slope of a curve at a single point: how fast it's rising or falling right there. You get it by taking the slope between two points and sliding them together.
Why tho? Three levels deeper
How can you get a slope from one point?
You can't directly, so you use two points a distance h apart: slope = (f(x + h) − f(x)) / h. Then you let h shrink toward 0 and see what number the slope settles on. That number is the derivative.
What does that give for y = x²?
(x + h)² − x² = 2xh + h², and dividing by h gives 2x + h. As h shrinks to 0, that becomes 2x. So the slope of y = x² at any point x is 2x.
Where does this show up in real life?
Speed is the derivative of position: a speedometer shows how fast your position is changing right now. Acceleration is the derivative of speed.
Average versus instant
Driving 120 km in 2 hours is an average speed of 60 km/h. The speedometer shows your speed at one instant. A derivative is that instant-speed idea, for any changing quantity.
The picture to keep in mind
A straight line through two points on a curve (a secant). Slide the points together until they meet, and the line just touches the curve (a tangent). The tangent's slope is the derivative.
Notation
You'll see it written f′(x) or dy/dx: "the change in y for a tiny change in x".
Check yourself
For y = x², what is the slope at x = 3?
Answer
6, from the slope formula 2x.