The directional derivative of $f\left(x\,y\right)$ or $f\left(x\,y\,z\right)$ is the rate of change of $f$ in a given direction specified by a unit vector u.

A common notation for the directional derivative at point P and in the direction u is ${\mathrm{D}}_{\mathbf{u}}f\left(P\right)$.

When the directional derivative is known to exist, it can be computed as the dot product of the vector u and the vector

Conceptually, the directional derivative is obtained as the rate of change of the values of $f$ along a line that is through point P and that has direction u. Since the line is parametrized by a single parameter such as $t$, the function values along this line are similarly parametrized, so the rate of change of

The Student MultivariateCalculus package contains the DirectionalDerivative command whose arguments can be the function $f$, a list of its variables, and a vector (given as a list of components) in the appropriate direction. In this form, the command will automatically normalize the vector and return the generic directional derivative for an arbitrary point.

Alternatively, to obtain the directional derivative at a specific point P:$\left(a\,b\,c\right)$, modify the list of variables by equating the list of variables to the list $\left[a\,b\,c\right]$.

For functions of two variables, this command can return a graph that purports to show the surface, a plane tangent to the surface, the direction vector, and the direction vector scaled and projected onto the tangent plane. This visualization is more readily obtained via the  tutor.

There is a DirectionalDiff command in the Student VectorCalculus package, and a DirectionalDiff command in the Physics:-Vectors package. The command in the VectorCalculus package will not automatically normalize the direction vector, and makes no provision for evaluation at a point. The command in the Physics:-Vectors package does not require normalization of the direction vector, but requires the complete machinery of vectors within this package. Neither of these two commands will be explored further.

With the Student MultivariateCalculus package loaded, the Context Panel, launched on an expression in two or three variables, provides interactive access to the DirectionalDerivative command.