Section 7.1 The Definition of the Directional Derivative

The translucent blue surface is the graph of \(z = f(x,y)\text{.}\) In the \(xy\)-plane below it, the point \(P_0(x_0,y_0)\) is fixed, the short blue arrow is the unit vector \(\mathbf u\text{,}\) and the gold arrow is the displacement \(s\mathbf u\text{,}\) which ends at the point \(P(x_0+su_1,\, y_0+su_2)\text{.}\) The dashed vertical lines rise from \(P_0\) and \(P\) to the surface, where the values \(f(P_0)\) and \(f(P)\) are marked. Drag the slider to change the distance \(s\text{,}\) or press “s \(\to\) 0” to animate the limit in the definition of the directional derivative: as \(s \to 0\text{,}\) the point \(P\) slides back toward \(P_0\) and the difference quotient \(\big(f(P) - f(P_0)\big)/s\text{,}\) displayed above the figure, approaches \(\left(D_{\mathbf u} f\right)_{P_0}\text{.}\) Drag the figure to view it from a different angle, or press “Rotate” to spin it.

Drag the slider for the angle to choose the unit vector \(\mathbf u\text{,}\) and drag the slider for \(s\) to move the point \(P\) along the direction \(\mathbf u\) starting at \(P_0(1,1)\text{,}\) for the function \(f(x,y) = 1 + x^2 + y^2\text{.}\) The readout compares the difference quotient \(\big(f(P) - f(P_0)\big)/s\) with its limit as \(s \to 0\text{,}\) which is the directional derivative \(\left(D_{\mathbf u} f\right)_{P_0}\text{.}\)