The opts argument can contain any of the following equations that set output options.

directionoptions = list

frames = posint

functionoptions = list

output = value, plot, or animation

output = value specifies that the computed value of the derivative is returned. Plot options and the range arguments r1, r2, and r3 are ignored if output = value.

output = plot specifies that a plot of the directional derivative is displayed, showing the function, the tangent plane at the specified point, the direction vector, and the direction vector projected onto the tangent plane at the specified point. Note that the length of the vector in the tangent plane corresponding to the specified direction is chosen to illustrate the concept and is not the magnitude of the directional derivative. Also, the plotting functionality is only available if f is a real-valued (scalar) function of two variables.

output = animation specifies that an animation is displayed, showing the function, the tangent plane, the direction vector (whose direction varies as the animation plays), and the direction vector projected onto the tangent plane at the specified point. Note that the animation functionality is only available if f is a real-valued (scalar) function of two variables.

showdirection = true or false

showfunction = true or false

showtangentplane = true or false

tangentoptions = list

For information on how to change the default colors, see the Student[SetColors] help page.

caption = anything

The DirectionalDerivative command returns the directional derivative of f, evaluated at pt if it is specified, in the direction given by dir; that is, the product of the Jacobian matrix of the function f, evaluated at pt if it is specified, and the normalized direction vector dir.

In particular, if f is a real-valued (scalar) function, then this command returns the dot product of the gradient of f and the normalized direction vector dir, which is a scalar.

If the output option in opts is specified to be plot or animation, the command returns a plot of the directional derivative or an animation of the directional derivative in various directions, respectively. Otherwise, by default, the output is the computed value of the directional derivative.

The DirectionalDerivativeTutor routine offers equivalent capabilities to DirectionalDerivative in a tutor interface. See the Student[MultivariateCalculus][DirectionalDerivativeTutor] help page.

For the DirectionalDerivative command, it is assumed the user is doing real variable calculus, however, complex numbers can be generated by some calculations.

$\mathrm{with}\left(\mathrm{Student}\left[\mathrm{MultivariateCalculus}\right]\right)\:$

$\mathrm{DirectionalDerivative}\left(x^2+y^2\,\left[x\,y\right]=\left[1\,2\right]\,\left[3\,4\right]\right)$

$\frac{22}{5}$

$\mathrm{DirectionalDerivative}\left(x^2+y^2\,\left[x\,y\right]=\left[-4\,4\right]\,\left[-4\,4\right]\,\mathrm{output}=\mathrm{animation}\,\mathrm{frames}=7\right)$

$\mathrm{DirectionalDerivative}\left(x^2+y^2\,\left[x\,y\right]=\left[-4\,4\right]\,\left[-6\,-6\right]\,x=-8..-2\,y=0..6\,z=0..40\,\mathrm{output}=\mathrm{plot}\right)$