Instructions.

Press “Play” to watch the whole construction in one continuous motion: the vertical plane through \(P_0(1,1)\) in the direction \(\mathbf u\) cuts the surface \(z = 1 + x^2 + y^2\) in the gold curve, the red secant joins \((P_0, f(P_0))\) to \((P, f(P))\text{,}\) and as \(s \to 0\) that secant rotates onto the green tangent line. The reading at the lower left compares the secant slope \(2\sqrt 2 + s\) with the tangent slope \(2\sqrt 2\text{,}\) which is the directional derivative \(\left(D_{\mathbf u} f\right)_{P_0}\text{.}\) Stop the animation at any time and drag the \(s\) slider yourself, or press “s \(\to\) 0” to replay just the limit. Drag the figure to change the viewpoint, or press “Spin” to let it turn.