We have previously described how an expression can be differentiated with respect to a varaible; however, we have also discussed function algebra. To differentiate a function, we will use the D operator.
[> D( cos );
$-sin$
[> D( sec + 2*csc + 3*cot + 4 );
$-3 \cot^2 - 2 \cot \csc + \sec \tan - 3$
[> id := (x) -> x;
$id := x \mapsto x$
[> p := id^3 + 5*id^2 + 2*id + 1;
$p := id^3 + 5 id^2 + 2 id + 1$
[> D( p );
$3 id^2 + 10 id + 2$
[> D( p )( 1 ); # Evaluate the derivative at 1
$15$
If Maple does not know anything about the function, it will return unevaluated:
[> D( sin + y + 1 );
$\cos + D(y)$
[> D( sin@(2*id) ); # diff( sin( 2*x ), x )
$2 \cos@(2 id)$
You can calculate higher derivatives by using D@@n to calculate the $n$th derivative:
[> (D@@4)( sin@(2*id) ); # diff( sin( 2*x ), x, x, x, x )
$16 \sin@(2 id)$
We will use unevaluated functions to write out differential equations and Taylor series:
[> D(y)(t) = y(t) + sin(t);
$D(y)(t) = y(t) + \sin(t)$
[> f(x + h) = f(x) + D(f)(x)*h + (D@@2)(f)(x)*h^2/2 + (D@@3)(f)(x)h^3/6;
$f(x + h) = f(x) + D(f)(x) h + \frac{1}{2} D^{(2)}(f)(x) h^2 + \frac{1}{6} D^{(3)}(f)(x) h^3$