Consider a point [math]\left(x_0,y_0\right)[/math] in the domain of [math]f\left(x,y\right)[/math]. Now suppose you "move through" [math]\left(x_0,y_0\right)[/math] in the direction indicated by some unit vector [math]\mathbf{u}=\left(u_1,u_2\right)[/math]. As you move through [math]\left(x_0,y_0\right)[/math] in the direction of [math]\mathbf{u}[/math], what is the rate of change of [math]f[/math]? This is the idea behind [i]the directional derivative[/i]. No longer are we constrained to move only in the positive [math]x[/math] or positive [math]y[/math] direction to find the rate of change of [math]f[/math].[br][br]The derivative of [math]f[/math] at the point [math]\left(x_0,y_0\right)[/math] in the direction of [math]\mathbf{u}[/math] is denoted [math]\left(D_{\mathbf{u}}f\right)_{\left(x_0,y_0\right)}[/math].[br][br]In the interactive figure, drag the end of the [math]\mathbf{u}[/math] vector to change it. Note the axis labels in the bottom-right pane; the horizontal axis indicates distance away from [math]\left(x_0,y_0\right)[/math] in the direction of [math]\mathbf{u}[/math].

[i]Developed for use with Thomas' Calculus, published by Pearson.[/i]