Instructions.

The surface \(z = 1 + x^2 + y^2\) is cut by the vertical plane through \(P_0(1,1)\) in the direction \(\mathbf u\text{.}\) Drag the slider to change \(s\text{:}\) the point \(P\) slides along \(\mathbf u\) in the \(xy\)-plane, and the secant line through the surface points \((1,\,1,\,f(P_0))\) and \((1+su_1,\,1+su_2,\,f(P))\) rotates toward the tangent line. Press “s \(\to\) 0” to animate the limit; the readout shows the secant slope \(2\sqrt 2 + s\) approaching the directional derivative \(2\sqrt 2\text{.}\) Drag the figure to view it from a different angle, or press the “Rotate” button.