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.

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

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

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

- 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 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)
- For each of the points from part c, find the equation of the line that is tangent to the graph of at these points. Plot the function and the tangent lines on the same graph over the interval given in part a to show the lines have the same slope.

- Given:

- a)
- Find the tangent plane at .
- b)
- Plot the function and the tangent plane on the same graph and rotate to see the point of tangency.
- c)
- Find the point on the graph of where the tangent plane is horizontal. Plot the function and this tangent plane on the same graph and rotate to see the point of tangency.

2011-09-10