Cannot differentiate by a defined symbolic function

Hello,

I am unclear about the behaviour of diff() and functionalDerivative() when the second argument is a defined function.

Snippet-1

syms x(t)
f = x^2 + 3*x;
Df = diff( f, x );                  % correctly produces: 2*x(t) + 3
Df2 = functionalDerivative( f, x ); % correctly produces: 2*x(t) + 3

Snippet-2

syms x(t)
x = symfun( t^2, t );
f = x^2 + 3*x;                      % evaluates to an expression in 't'
Df = diff( f, x );                  % error: "Arguments, except for first, must not be defined functions"
Df2 = functionalDerivative( f, x ); % also error? "Second argument must be a variable, [or] a symfun .."

In this case, x(t) is infact a symfun object -- and I might have expected a partial derivative expression.

Defining f before x (such that f remains an expression in x) has no effect either. Am I missing something? Why must the second argument not be defined, and why does functionalDerivative not produce 2*x(t) + 3?

Help appreciated! Thanks, and apologies if I am mistaken here.

Answer 1

[Shubham Rawat]

Hi Ajay,

In those line it is showing error as x should be a variable symfunction not a defined sym function. Here sym function 'x' is defined.

If you want to calculate differentiation you can do like this

Df = diff(f, t);
Df2 = functionalDerivative( f, x ); %but this won't work

Definition of Functional Derivative

A functional derivative is the derivative of a functional with respect to the function that the functional depends on.

You may reproduce your code by:

syms x(t)
f = x^2 + 3*x;                     
Df = diff( f, x );                
Df2 = functionalDerivative(f, x);

You can see the documentation for furthere information:

symfun differentiation functional Derivative

Hope this Helps!