Directional derivative and gradient


Scroll down to the bottom to view the interactive graph.


The directional derivative of a function \(f(x,y)\) at a point \(P_0 (x_0,y_0)\) in the direction \(\textbf{u} = \left\langle a,b \right\rangle\), denoted by \[D_{\textbf{u}} f(x_0,y_0),\] is recognized as the slope of the line tangent to the surface \(z = f(x,y)\) at \(P_0\) in the direction \(\textbf{u}\) on the domain.

As long as \(f\) is differentiable at \(P_0\), we can use the tangent plane to come up with a simple formula \(D_{\textbf{u}} f = \textbf{u} \cdot \left\langle f_x,f_y \right\rangle.\) This directional derivative is at its maximum value if \(\textbf{u}\) points to the direction \(\nabla f = \left\langle f_x,f_y \right\rangle\).

If \(\textbf{u}\) points to the direction tangent to a level curve, then we must have \(D_{\textbf{u}} f = 0\). This also holds when \(\textbf{u} \perp \nabla f\). As a result, the vector \(\nabla f\) points to the direction perpendicular to a level curve.

The sign of \(D_{\textbf{u}} f\) can be determined based on the angle \(\theta\) between \(\textbf{u}\) and \(\nabla f\). The derivative \(D_{\textbf{u}} f\) is positive if \(\theta \in [0,\pi/2)\) and is negative if \(\theta \in (\pi/2,\pi]\).


To view the interactive graph:

  1. Make sure you have the latest version of Java 7 installed in your computer. Tablets and smartphones are not supported.
  2. Download this file, directionalderivative.ggb.
  3. Click here to open GeoGebra.
  4. After you open GeoGebra, click "File" in the toolbar, then click "Open".
  5. Choose the .ggb file you just downloaded and click the "Open" button.
  6. Now you should be able to view the graph inside GeoGebra.

To rotate the graph, right click and drag.

Joseph Lo