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); > diff(f(x,y),x,x); > diff(f(x,y),y,y); > diff(f(x,y),x,y); > D[1](f)(x,y); > D[2](f)(x,y); > D[1,1](f)(x,y); > D[2,2](f)(x,y); > D[1,2](f)(x,y);

The next example shows how to evaluate the mixed partial derivative of the function given above at the point .

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

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

- Given the single variable function
- a)
- Plot over the interval .
- b)
- Find the slope of at .
- c)
- Find the two values that have the same slope as in part b.
- d)
- Find the slope of the two secant lines containing the value from part b and each of the two x values in part c.
- e)
- Find the equation of a single line that is tangent to the graph of at two points: the point from part b and one from part c. Plot the function and the tangent line on the same graph over the interval given in part a to show the line is tangent at both points.

- Given:

- a)
- Find the tangent plane at .
- b)
- Plot the function and the tangent plane over the intervals and and rotate the plot so that you can see that the plane is tangent.

- There is only one point at which the plane tangent to the surface is horizontal. Find it and plot it along with the function on the same graph. Be sure to use axes so that you can rotate the graph and see that the tangent plane is horizontal.

2008-09-15