What a derivative is
The slope of a curve at one point, found by letting a secant shrink.
A straight line has one slope. A curve has a different steepness at every point, and the derivative is the name for that steepness at one point.
You cannot measure a slope at a single point: slope needs two points. So take a second point a small step h away and draw the line through both. That line is a secant, and its slope is the change in f divided by h.
Now let h shrink. The secant turns, and its slope settles on one number. That number is the derivative f′(a): the slope of the tangent at a.
For f(x) = x² the slope of the secant from a to a + h works out to 2a + h. Letting h go to 0 leaves 2a, which is the rule you may know: the derivative of x² is 2x.
A derivative is a rate: how fast the output changes for a small change of the input. Velocity is the derivative of position; marginal cost is the derivative of cost.
Answer it first
The secant slope of f(x) = x³ from 1 to 1 + h is 3 + 3h + h². What is f′(1)?
Show the answer
3. As h goes to 0 the terms 3h and h² vanish. It agrees with the rule 3x² at x = 1.
Reading is not learning it
Leo teaches this at your pace, asks for your attempt before he shows his, and checks a week later what stayed.