Read it again any time, including while you work through the exercises.
On a straight line, the slope is the same everywhere: it is the gradient. But on a curve, the slope keeps changing! Think of a mountain road: sometimes it climbs steeply, sometimes it is almost flat, sometimes it goes down. How can we measure the slope of a curve at one precise point? That is exactly the job of the derivative.
First idea: between two points of a curve, we can compute an average slope, called the rate of change: (f(b) - f(a)) ÷ (b - a). It is the slope of the line joining the two points, like the average speed of a journey.
Second idea, the stroke of genius: bring the two points closer. We compute the rate of change between a and a + h, then let h tend to 0. If this rate approaches a fixed number, that number is the derivative of f at a, written f'(a). It is the instantaneous slope, like the speed shown on the dashboard at one precise moment.
Geometrically, f'(a) is the slope of the tangent line: the line that skims the curve at the point with x-coordinate a, matching its direction. Its equation is y = f'(a)(x - a) + f(a). A useful special case: if the tangent is horizontal, then f'(a) = 0, which often happens at the top of a bump or the bottom of a dip.
For the function f(x) = x², we will accept the result computed in the example: f'(x) = 2x. At the point with x-coordinate 3, the slope of the curve is therefore 2 × 3 = 6.