Higher Order Derivatives

The derivative represents the slope of a function. It can also represent the instantaneous velocity. The second derivative represents the acceleration and can also represent the concavity of a function. In order to calculate these and any other higher order derivative in Maple the commands D and diff are used. For the function $f(x)=7x-\sin^3(x)$ the second and third derivative will be calculated using both commands.

> f:=x->7*x^3-sin(x)^3;
> plot(f(x), x=-2..2);
> (D@@2)(f)(x);
> D[1,1](f)(x);
> diff(f(x),x,x);
> diff(f(x),x$2);

The 2 in the first D command stands for the second derivative. The 1 in the second D command means ``with respect to x''.

> (D@@3)(f)(x);
> D[1,1,1](f)(x);
> diff(f(x),x,x,x);
> diff(f(x),x$3);

In order to substitute an $x$ value into the higher order derivative the subs command must be used for diff; but for D the $x$ value can be incorporated into the command itself.

> evalf(subs(x=0,diff(f(x),x,x,x)));
> (D@@3)(f)(0);
> D[1,1,1}(f)(0);

Remember from your work with the first derivative that the D command cannot be used with expressions.

> g:=19*x^5-14*x+100;
> D(g)(x);
> diff(g,x);