Derivatives: From Secant to Tangent
The slope of a curve at a point is what the slope of a chord becomes as its two ends squeeze together. Shrink h and watch the secant turn into the tangent.
The idea of a limit
A straight line through two points on a curve, (a, f(a)) and (a + h, f(a + h)), is a secant. Its slope is the difference quotient [f(a + h) − f(a)] / h. You can't just put h = 0 (that gives 0/0), but you can see where the slopes are heading as h shrinks. That limiting value is the derivative f′(a), the slope of the tangent.
The derivative is a function too
Do this at every point and you get a new graph, f′(x). Where f rises, f′ is positive; where f has a peak or valley, f′ = 0. Some functions have a sharp corner, like |x| at 0: the slope from the right (+1) and from the left (−1) disagree, so there is no derivative there. Try a negative h to approach from the left.