Derivative Builder CalcLens

Slide a point along f and read the slope of the tangent there. Each slope is dropped onto the second graph as a height. Sweep all the way across and the derivative draws itself — f′ is not a new kind of object, it is a graph of slopes.

Try x^3-3x, sin(x), x^2, abs(x), e^x, ln(x).

Top: f, with the secant and the tangent at the point. Drag anywhere on this graph to move the point.

Bottom: every slope you have visited, plotted as a height.

What the numbers say

Derivative Builder

The top graph shows f. A point slides along it, carrying a tangent line. The slope of that tangent is plotted on the bottom graph as a height. Sweep across and the bottom graph fills in: that is f′.

The secant gap h

The dashed secant joins . Its slope is the difference quotient . Drag h toward 0 and watch the secant settle onto the tangent, and the difference quotient settle onto f′(x). That limit is the definition.

What to watch for

Keyboard

← →
Move the point (focus the x slider first)
Space
Start or stop the sweep
?
This dialog
Esc
Close