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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.