From secant to tangent: what a derivative is
Slide a point Q along the graph towards P, watch the gradient of the secant in a table settle on the gradient of the tangent, draw the tangent, and see why a corner has no derivative.
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Guide
A secant joins two points P and Q of the graph. Its gradient is the average rate of change f(x + h) − f(x)h. As Q slides towards P (h → 0), the secant turns into the tangent.
The derivative at a point is the limit of the gradients of the secants: f′(x) = lim(h→0)f(x + h) − f(x)h. That is the gradient of the tangent at the point.
If the graph has a corner at a point, the gradient from the left and from the right differ, and there is no derivative.