Derivatives, from first principles
Five steps from “a function is a rule” to f′(x) = 2x, derived by hand and then checked against a measurement.
Step 1 A function is a rule.
Now you: drag the point
The tangent turns as you move, the working shows your numbers substituted in, and every slope you visit is plotted below until the slopes make a curve of their own.
The working, with your numbers in it
f(x) = x³ − 3x at x = 0.80
f(0.80) = -1.89the height of the curve there
slope = rise ÷ run = -0.54 ÷ 0.50 = -1.08
f′(x) = 3x² − 3 → exact -1.08the measured slope differs by 1.0e-6
The slope hits zero twice, at x = −1 and x = 1. Those are the two turning points, and they are where the derivative curve crosses its own axis.
The point snaps to steps of 0.05, so every number here can be checked on a calculator: -0.54 ÷ 0.50 gives exactly what the panel says.