Tangents and the derivative

The gradient of a curve at a point is the derivative. Drag the point along the curve, shrink the secant step h towards zero, and switch on the f′(x) graph to see where the gradient is zero, positive or negative.

Predict first

Choose f(x) = x² with the point at x = 1, where the gradient is 2. You drag the point to x = 2. What is the gradient there?

f(x) tangent secant through x and x + h f′(x). Drag on the graph to move the point.

Function

Preset

Point, x

Secant step, h

Show secant (limit as h → 0) Show derivative graph f′(x)

Try this

Choose 'Limit demo (h → 0)' and reduce h from 2.00 to 0.02. How does the secant gradient compare with f′(1) = 2, and how large is the difference at h = 1?
Choose 'Stationary points' and drag the point to where the tangent is horizontal. What value does f′ read at x = −1 and at x = 1, and what happens to the sign of f′ on either side?
Choose 'sin → cos' and move the point to x = 3.15 (about π). What is the gradient of the tangent there, and where on the f′ graph does that point sit?

Exam tip: where f′(x) = 0 there is a stationary point, and the sign of f′ either side tells you whether the curve is increasing or decreasing, so quote the sign change when you justify a maximum or minimum.