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Calculus, the big ideas

Calculus sounds scary, but it's built on two simple questions: how fast is something changing? and how much adds up over time? That's it.

First, limits

A limit asks: as I get closer and closer to some point, what value am I heading toward? You don't have to reach it — you just look at where you're going.

Limits matter because they let us talk about an exact instant — not an average over a stretch, but the value at a single point. That idea unlocks everything else.

Derivatives: how fast it's changing

Picture driving. Your average speed is distance ÷ time. But your speedometer shows your speed right now — that's a derivative: the rate of change at one instant.

On a graph, the derivative is the slope of the curve at a point. The most common shortcut is the power rule:

(The little apostrophe f'(x) just means "the derivative of f".)

Integrals: how much adds up

The integral is the reverse idea: instead of the rate of change, it's the total accumulated. On a graph, it's the area under the curve.

Back to the car: if the derivative turned distance into speed, the integral turns speed back into distance. Knowing how fast you went and for how long, you can add it all up to get the total distance traveled.

The big connection

Derivatives and integrals are opposites — one undoes the other. That's the heart of the Fundamental Theorem of Calculus: breaking things apart to see the instant rate, and adding them back up to see the whole.