Techniques of differentiation

Understand limits and derivatives · Diagrams

Let a secant approach a tangent

Move a secant towards the tangent

Illustrative example: y = x² with P = (1, 1). Move Q = (1 + h, (1 + h)²) towards P. The source sketch uses a general smooth curve, not this particular formula.

Illustrative curve y = x²

Size of h

More precision

Approach from

Points on the curve
Point	x	f(x)
P · fixed	1	1
Q · moving	1.1	1.21

The dashed line is the tangent at P; the segment PQ is the secant.

h = 0.1. Secant gradient = 2.1; tangent gradient = 2.

Values are rounded to 9 decimal places. Use the table to inspect points that overlap on the graph.

Nearby values illustrate the limit; a finite table alone is not a proof.

Teaching and source notes

The source sketch is a general smooth curve with P(x, y), Q(x + h, y + Δy), secant PQ and tangent AB. This numerical example is explicitly y = x² at P = (1, 1), not a recovered formula for the original drawing.

For h ≠ 0, Δy = (1 + h)² − 1 = h(2 + h), so Δy/h = 2 + h. Its limit is 2 from either side. At h = 0 the difference quotient is undefined; the derivative comes from its limit. A secant need not tend to a tangent where the derivative does not exist.

An illustrative smooth curve shows P(x,y), Q(x+h,y+Δy), secant PQ and tangent AB. The difference quotient Δy/h tends to the tangent gradient as h approaches 0 if the derivative exists.

Open original diagram

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