The Maple commands for computing partial derivatives are `D`
and `diff`. The `diff` command can be used on both expressions and functions whereas the `D` command can be used only on functions. The examples below show all first order and second order partials in Maple.

> f := (x,y) -> x^2*y^2-x*y; > diff(f(x,y),x); > diff(f(x,y),y,y); > D[1](f)(x,y); > D[1,2](f)(x,y);Note in the above D command that the

> subs({x=-1,y=1},diff(f(x,y),x,y)); > D[1,2](f)(-1,1);

> g := x-> sin(x)-x^3/7+x^2; > tl := D(g)(5)*(x-5)+g(5); > plot({g(x),tl},x=-2..8);

The next example shows how to find the tangent plane to the function
at . You could write the partials with `diff` or `D`. This example uses `D` as it is easier to plug in the the point with this syntax; with `diff` the `subs` command would be used.

> f:=(x,y)->1/(1+x^2+y^2); > tp:=D[1](f)(1/8,1/4)*(x-1/8)+D[2](f)(1/8,1/4)*(y-1/4)+f(1/8,1/4); > plot3d({f(x,y),tp},x=-1..1,y=-1..1,style=patchnogrid);

To find a point where the tangent plane is horizontal, you would need to solve where both first order partials are equal to zero simultaneously.

> solve({diff(f(x,y),x)=0,diff(f(x,y),y)=0},{x,y});

There are two plot commands for three-dimensional graphs **plot3d** and **implicitplot3d**. The first assumes the **z=f(x,y)** and the **z** is therefore not included in the command.

>plot3d(g(x,y),x=-10..10,y=-10..10,axes=boxed);If

>with(plots): >surf:=x^2+y^2=1-z^2; >implicitplot3d([surf,x=y],x=-1.1..1.1,y=-1.1..1.1,z=-1.1..1.1,axes=normal,color=[black,magenta],style=[wireframe,patchnogrid],thickness=2);Three-dimensional plots have many options. Some have been used in the above command. To see more information try these two commands.

>?plot,colornames >?plot3d,options

- Compute the three distinct second order partial derivatives of

at the point using the**diff**command and then again using the**D**command. - Given the function

- a)
- Plot the function and the plane on the same graph. Use intervals , , .
- b)
- Find the derivative of in the plane.
- c)
- Graph the two-dimensional intersection of the plane and .
- d)
- Does your two-dimensional graph look like the intersection from your three-dimensional graph? Be sure to use the same ranges to properly compare and rotate the 3-D graph.

- Find the equation of the plane tangent to the surface at the point and find the equation of the plane tangent to the graph at . Plot both tangent planes on the same graph as the surface over the intervals and . Be sure to rotate the graph to see that the planes are tangent to the surface.

2015-09-08