How do I symbolically work with implicit functions (i.e., "x(t)") within the Symbolic Math Toolbox?

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

[MathWorks Support Team]

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(' ')

[Walter Roberson]

From R2015a onwards, there is a new functionalDerivative() that would allow you to take the derivative with respect to a function.

[Walter Roberson]

From about R2011b onwards, instead of using

syms t
x = sym('x(t)');

you can also use

syms t x(t)

or even

syms x(t)

which will notice if t is not already defined as a symbol and will define it as a symbol if it needs to.