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From: Joe Riel <joer@san.rr.com>
Newsgroups: comp.soft-sys.math.maple
Subject: Re: Differentiating with respect to an expression
Date: Tue, 11 Nov 2014 14:35:07 -0800
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rouben@shadow.(none) (Rouben Rostamian) writes:

> The following issue comes up quite often in the context of
> analytical mechanics.  I have a clunky solution for it.
> I wonder if there is a clever way.
>
> Let's say we have L = x^2 + x'^2 * x''^2, where x is a function of t,
> and in the usual mathematical notation, x' and x'' are the first and
> second derivatives of x.
>
> We want to compute the derivative of L with respect to x'.  The
> answer should be 2*x'*x''^2.  Here is the way I do it in Maple:
>
> restart;
> L := x(t)^2 + diff(x(t),t)^2 * diff(x(t),t,t)^2;
> subs(diff(x(t),t,t)=Z2, diff(x(t),t)=Z1, %);
> diff(%, Z1);
> subs(Z1=diff(x(t),t), Z2=diff(x(t),t,t), %);
>
> The L shown above is simple enough so that we don't need a
> CAS to compute the derivative.  The L in a real example will
> be the result of a long chain of calculations, will depend on
> several functions and their derivatives, and will take up a
> couple of screenfuls.
>
> If there is a clever way to compute that derivative, I would
> like to know.

A low-level way to do this is with frontend:

frontend(diff, [L,diff(x(t),t)]);
       2*diff(x(t),t)*diff(diff(x(t),t),t)^2

An alternative method is to use the Physics package:

with(Physics):

diff(L, diff(x(t),t));
       2*diff(x(t),t)*diff(diff(x(t),t),t)^2

# alternatively, with Physics

diff(L, D(x)(t));
     2*D(x)(t)*`@@`(D,2)(x)(t)^2


-- 
Joe Riel