Secant to tangent – the derivative as a limit

What it shows

The gradient of the secant through P(a, f(a)) and Q(a + h, f(a + h)) is the average rate of change (f(a + h) − f(a))/h. As h shrinks towards 0 the secant turns into the tangent and the quotient approaches the derivative f′(a), the instantaneous rate of change; the first-principles algebra for x² and x³ shows why. Zooming in on P reveals local linearity: near P the curve looks like its linearisation L(x). At a corner, a cusp or a vertical tangent the one-sided limits differ or are infinite, so f′(a) does not exist. The ball-throw graph turns gradients into velocities in m/s.

How to use

Choose a Function, then drag P and Q along the curve or move the h slider. Press Animate h → 0 and watch the secant swing onto the tangent while the table of difference quotients settles on f′(a). Tick Tangent and Normal to see both lines and their equations, and raise Zoom on P to test local linearity. Try |x|, ∛(x²) and ∛x with P at 0.

Parameters you can change

• Function f(x) = x², f(x) = x³, f(x) = sin x, f(x) = eˣ, f(x) = |x| (corner), f(x) = ∛(x²) (cusp), f(x) = ∛x (vertical tangent), Ball thrown up: s(t) = 20t − 4.905t²
• x-coordinate a of P -3–4
• Step h (Q is at a + h) -2–2
• Show tangent
• Show normal
• Zoom on P (×10^zoom) 0–3

Questions to explore

1. For f(x) = x² at a = 1, what value do the quotients in the table approach as h gets smaller?
2. Why do the quotients for |x| at 0 approach different values for positive and negative h?
3. At what time is the ball's instantaneous velocity zero, and what does the tangent look like then?