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As new features are being built into upcoming releases of Maple, here is one request that would be very helpful for those of us who use Maple to teach lower-level (Calculus) students. Maple can work nicely with functions, but students are not always so comfortable with this language. Here's a current example. Suppose you want to find the tangent line to a function. We might work as follows. 
> f := x -> sqrt(4-x^2);
                                      /     2\
                             x -> sqrt\4 - x /
> df := D( f );           # typed, or from context-menu
                                         x      
                            x -> - ------------
                                       /     2\
                                   sqrt\4 - x /
> TL := f(1)+df(1)*(x-1); # tangent line at (1,sqrt(3))
                           (1/2)   1  (1/2)        
                          3      - - 3      (x - 1)
                                   3
This works nicely and looks fine. Now change the function. 
f := x -> sin(x);
                                 x -> sin(x)
df := D( f );           # typed, or from context-menu
                                     cos
TL := f(1)+df(1)*(x-1); # tangent line at (1,sin(1))
                           sin(1) + cos(1) (x - 1)
This works fine, but does not LOOK good. The problem is that Maple is TOO SMART. It knows that the derivative function (x->cos(x)) is simply the cosine FUNCTION. Instead of the abbreviation (cos) I want to be able to tell Maple to show this result as (x->cos(x)). The problem is that this output is not something that students would have seen previously. This discrepancy in the appearance of the output occurs only when the derivative is so simple that Maple recognizes it as one of its built-in functions. Some slight changes to the problem allow us to avoid this annoyance. 
f := x -> 2*sin(x);
                               x -> 2 sin(x)
df := D( f );           # typed, or from context-menu
                               x -> 2 cos(x)
TL := f(1)+df(1)*(x-1); # tangent line at (1,cos(1))
                         2 sin(1) + 2 cos(1) (x - 1)
I have to admit that the previous example did surprise me. I thought Maple would be smart enough to recognize f as 2*sin and df as 2*cos, but we see otherwise. As a final example, consider 
f := x -> cos(x);
                                  x -> cos(x)
df := D( f );           # typed, or from context-menu
                                  x -> -sin(x)
TL := f(1)+df(1)*(x-1); # tangent line at (1,cos(1))
                           cos(1) - sin(1) (x - 1)
Here, I have no complaints but I am curious. Why didn't Maple recognize f as cos and produce a display that I could complain about. This example illustrates to me that Maple can be given the smarts to decide when NOT to simplify a function. Could this be solved be having a user-controlled setting of some sort? (Maybe something like _ShowFunctionDefn:=explicit;) It's another example of some user control that could make Maple much more effective for teaching. Thanks for listening, Doug
derivative

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