MATLAB: How to symbolically work with implicit functions (i.e., “x(t)”) within the Symbolic Math Toolbox

Question

I would like to find the derivative of:

   E1 =  x' * sin(x)

where "x" and "y" are functions of time "t".

I tried:

syms x  t
Dx= diff(x,t)

I receive the result:

 Dx =
 0

In addition

syms x y t
Dx= diff(x(t),t)

 ??? Unable to find subsindex function for class sym.

Answer 1

The following example demonstrates how you can represent "x" as functions of time "t" and perform differentation on it:

Example 1:

syms  t
x = sym('x(t)'); % x(t)
E1 = diff(x) * sin(x) 
E2 = diff(E1,t)
pretty(E1)
pretty(E2)

The solution for E1:

                               /d      \
                               |-- x(t)| sin(x(t))
                               \dt     /

The solution for E2:

                   / 2      \
                   |d       |             /d      \2
                   |--- x(t)| sin(x(t)) + |-- x(t)|  cos(x(t))
                   |  2     |             \dt     /
                   \dt      /

The following example demonstrates how to substitute 'x(t)' with an explicit definition.

Example 2:

% Order of the derivative
n = 2
% Define symbolic variables.
syms t
x = sym('t^2');
f = sym('f(x)');
% SUBS is used to substitute definition of x
f_subs = subs(f);
disp('Diff f_subs wrt to t')
pretty(diff(5*f_subs + t^2,t,n))
disp(' ')
disp('Diff f_subs wrt to x(t)')
pretty(diff(5*f_subs + t^2,x,n))
disp(' ')